KMS States on Nica-Toeplitz -algebraslink.springer.com/content/pdf/10.1007/s00220-020-03711-6.pdfKMS...

Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-020-03711-6Commun. Math. Phys. 378, 1875–1929 (2020) Communications in

MathematicalPhysics

KMS States on Nica-Toeplitz C∗-algebras

Zahra Afsar1, Nadia S. Larsen2, Sergey Neshveyev2

1 School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia.E-mail: [email protected]

2 Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway.E-mail: [email protected]; [email protected]

Received: 29 May 2019 / Accepted: 15 January 2020Published online: 3 March 2020 – © The Author(s) 2020

Abstract: Given a quasi-lattice ordered group (G, P) and a compactly aligned prod-uct system X of essential C∗-correspondences over the monoid P , we show that thereis a bijection between the gauge-invariant KMSβ -states on the Nica-Toeplitz algebraNT (X) of X with respect to a gauge-type dynamics, on one side, and the tracial stateson the coefficient algebra A satisfying a system (in general infinite) of inequalities, onthe other. This strengthens and generalizes a number of results in the literature in severaldirections: we do not make any extra assumptions on P and X , and our result can, inprinciple, be used to study KMS-states at any finite inverse temperature β. Under fairlygeneral additional assumptions we show that there is a critical inverse temperature βcsuch that for β > βc all KMSβ -states are of Gibbs type, hence gauge-invariant, inwhich case we have a complete classification of KMSβ -states in terms of tracial stateson A, while at β = βc we have a phase transition manifesting itself in the appearanceof KMSβ -states that are not of Gibbs type. In the case of right-angled Artin monoidswe show also that our system of inequalities for traces on A can be reduced to a muchsmaller system, a finite one when the monoid is finitely generated. Most of our resultsgeneralize to arbitrary quasi-free dynamics on NT (X).

Introduction

Given a C∗-algebra A with a time evolution σ = (σt )t∈R, the Kubo–Martin–Schwinger(KMS) condition on states of A gives a rigorous mathematical characterization of equi-librium of the system (A, σ ) at inverse temperatures β ∈ R. For example, in the case ofa full matrix algebra and the time evolution σt = Ad eit H , there is a unique KMSβ -statefor every β ∈ R, which is precisely the Gibbs state Tr(· e−βH )/Tr(e−βH ). Pimsner’sconstruction [27] of C∗-algebras T (X) and O(X) associated to a C∗-correspondenceX provided a new foundation for studying large classes of C∗-algebras in a systematic

Z.A. was supported by grants DP150101595 and DP170101821 of the Australian Research Council.

http://crossmark.crossref.org/dialog/?doi=10.1007/s00220-020-03711-6&domain=pdf

1876 Z. Afsar, N. S. Larsen, S. Neshveyev

way. The new framework proved fruitful also for studying KMS-states. After a numberof examples of systems (A, σ ) had been analyzed, with A a Cuntz-Pimsner algebraO(X) or a Toeplitz-Pimsner algebra T (X), and σ a quasi-free dynamics, a theory ofKMS-states for such systems was developed by Pinzari–Watatani–Yonetani [28] for fullfinite rank correspondences and the gauge dynamics and then by Laca–Neshveyev [21]for essential C∗-correspondences and general quasi-free dynamics.

In recent years there has been a growing interest in the structure of KMS-states onC∗-algebras associated with product systems of C∗-correspondences over quasi-latticeordered monoids, which generalize Cuntz-Pimsner and Toeplitz-Pimsner algebras. Theinitial motivation came from the work of Laca–Raeburn [22] on the semigroup C∗-algebra of the monoid Z+ � Z

×+ . The structure of KMS-states on this C∗-algebra, re-

vealed in [22], was impossible to explain using general results available at that time.Although such an explanation was soon provided by the groupoid picture [24], it wasstill natural to try to understand it from the point of view of C∗-correspondences. Thisled Hong–Larsen–Szymanski [14] and Brownlowe–an Huef–Laca–Raeburn [3] to un-dertake studies of KMS-states on C∗-algebras of product systems. The approach in [14]was in principle guided by [21], although in the end it employed more direct methodsusing special and quite strong hypotheses imposed on the correspondences, involvingexistence of orthonormal bases. A relaxation of the conditions on the correspondencesto existence of special Parseval frames enabled Afsar–an Huef–Raeburn [1] to studyequilibrium states on C∗-algebras of product systems over Z

n+ constructed out of fami-

lies of ∗-commuting local homeomorphisms on a compact Hausdorff space. In a relateddirection, Kakariadis [16] initiated a program to study KMS-states on Pimsner-type C∗-algebras arising from multivariable dynamics. C∗-algebras associated to k-graphs areanother large class of examples closely related to product systems whose structure ofKMS-states has been extensively studied in the recent years starting with [15]. Very re-cently, the analysis of KMS-states on the Toeplitz algebras of finite k-graphs has attaineda very satisfactory form in Christensen’s work [7] using the groupoid approach. Inter-esting results on KMS-states of the Toeplitz algebras of quasi-lattice ordered monoids,which can be considered as C∗-algebras associated with the simplest rank one productsystems, have been obtained in the work of Bruce–Laca–Ramagge–Sims [4].

This project started from the natural question whether it is possible to extend thetheory developed in [21] to algebras associated with product systems. The payoff ofour effort is a powerful machinery that clarifies and strengthens the existing results and,hopefully, can guide future investigations.

In order to formulate our motivation and results more precisely, let us first give somedetails of the theory developed in [21] in the case of the gauge dynamics. Assume we aregiven an essential C∗-correspondence X over a C∗-algebra A. The induction of traces viaX defines a map FX on the positive traces on A, which is an analogue of the dual Ruelletransfer operator in dynamical systems. Then for every β ∈ R, the map φ �→ φ|A definesa bijection between the KMSβ -states on T (X) with respect to the gauge dynamics andthe tracial states τ on A such that

e−βFXτ ≤ τ. (∗)As was shown in [21], similarly to the potential theory in classical probability, anytrace τ satisfying (*) decomposes into a finite part τ f = ∑∞

n=0 e−nβFn

Xτ0 (“potential”)and an infinite part τ∞ such that e−βFXτ∞ = τ∞ (“harmonic element”). This can alsobe thought of as an analogue of the Wold decomposition for isometries. The KMSβ -states corresponding to traces of finite type are easy to understand. They are obtained by

KMS States on Nica-Toeplitz C∗-algebras 1877

restriction from generalized Gibbs states on L(F(X)), which are constructed similarlyto the classical Gibbs states on full matrix algebras using induced traces on L(F(X)).Therefore the main content of the above description of KMS-states is the statement thatthe KMSβ -states of infinite type, that is, those without a Gibbs-type component, are ina one-to-one correspondence with the tracial states such that e−βFXτ = τ .

Now suppose that X = (X p)p∈P is a product system of C∗-correspondences over aquasi-lattice ordered monoid P . Then we obtain a collection of generalized dual transferoperators Fp, one for every correspondence X p. Assuming that X is compactly aligned,we consider the Nica-Toeplitz C∗-algebraNT (X) associated with X and the gauge-typetime evolution σ on it defined by a homomorphism N : P → (0,+∞). Similarly to thecase of a single correspondence, it is not difficult to see then that every tracial state τ onthe coefficient algebra A of the form τ = ∑

p∈P N (p)−βFpτ0 extends to a Gibbs-typestate onNT (X). Therefore the starting point of our investigation can be phrased as thefollowing questions:

(1) Is it possible to characterize the tracial states on A that are obtained from the KMS-states on NT (X) having no Gibbs-type components?

(2) Are there conditions that guarantee that all KMS-states are of Gibbs type? Whencan other KMS-states appear?

In this paper we will give a complete answer to question (1) and obtain quite generalresults answering the questions in (2).

We now describe in more detail the contents of the paper. In Sect. 1 we collect a num-ber of general results and definitions on quasi-lattice ordered groups, product systemsand their associated C∗-algebras, KMS-states and induced traces, and generalized dualRuelle transfer operators.

In Sect. 2 we consider a system (NT (X), σ ) as described above and show that anyKMS-state on NT (X), when restricted to the coefficient algebra A, gives a trace thatsatisfies a system of inequalities, see Theorem 2.1. We then go on to discuss the gener-alized Gibbs states and explain why our system of inequalities does not give interestinginformation for them.

In Sect. 3 we consider the free abelian monoids Zn+ for n ≥ 1 and show that any trace

on the coefficient algebra A satisfying the inequalities from Sect. 2 can be extended toa gauge-invariant KMS-state on NT (X). The reader interested in general quasi-latticeordered monoids can safely skip this section, but there are a number of reasons to treatthe monoids Z

n+ separately: this is a much studied case in the literature, see [1,15–17]

(as well as [3,14], where the monoid is ⊕∞k=1Z+, identified with Z

×+ ), and at the same

time it allows for a very short proof based on a perturbation trick from [21].In Sect. 4 we prove an auxiliary result on positivity of functionals on the core sub-

algebra of NT (X). The result is formulated and proved in a more abstract setting ofquasi-lattice graded C∗-algebras, which might be useful in other contexts.

In Sect. 5 we prove that a trace satisfying the inequalities from Sect. 2 can be uniquelyextended to a gauge-invariant KMS-state onNT (X), nowwithout any restrictions on P .The proof relies on the results of Sect. 4, but otherwise follows the familiar strategy of firstextending a trace to the core subalgebra and then using the gauge-invariant conditionalexpectation to extend the trace to a state on the entire algebraNT (X). This way we geta classification of the gauge-invariant KMS-states on NT (X) in terms of traces on A,which is the first main result of the paper, see Theorem 5.1.

In Sect. 6we show that everyKMS-state onNT (X) uniquely decomposes into a finiteand an infinite part, or in otherwords, into aGibbs-type functional and a functionalwhichdoes not dominate any functional of Gibbs type, see Proposition 6.6 and Theorem 6.8.


We show then that whether a KMS-state is of finite or of infinite type is determinedcompletely by its restriction to the coefficient algebra A, see Corollary 6.10. Togetherwith our general characterization of the restrictions of KMS-states to A this gives acomplete answer to question (1) above.

In Sect. 7 we prove that if the restriction τ of a KMSβ -state φ on NT (X) is suchthat the trace

∑p∈P N (p)−βFpτ is finite, then φ is of Gibbs type, see Theorem 7.1. In

particular, if the series∑

p∈P N (p)−βFpτ are convergent for all tracial states τ , thenall KMSβ -states are of Gibbs type and we get a complete classification of KMSβ -statesin terms of traces on A, see Corollaries 7.3 and 7.5. This gives an answer to the firstquestion in (2) and is the second main result of the paper.

These results confirm the general principle, known to many researchers who havestudied KMS-states, though perhaps not directly formalized in print, saying that theKMSβ -states for β in the region of convergence of a suitably defined zeta-function ofthe time evolution are of Gibbs type. There are many results of this sort in the literature,proved in different contexts, and usually requiring quite strong assumptions, see, e.g.,[1, Theorem 6.1], [14, Theorem 4.10], [15, Theorem 6.1], [20, Proposition 1.2], [22,Theorem 7.1]. By contrast we see that in our setting of product systems over quasi-lattice ordered monoids this principle is valid in great generality.

When N (p) ≥ 1 for all p, there is a smallest number βc such that the series∑p∈P N (p)−βFpτ are convergent for all positive traces τ and β > βc. We show

that if P is finitely generated and suitable additional assumptions are satisfied, there is aKMSβc -state which is not of Gibbs type, see Proposition 7.7. Therefore βc is the largestcritical inverse temperature at which we have a phase transition. This gives an answerto the second question in (2).

Section 8 is inspired by the recent work of Bruce–Laca–Ramagge–Sims [4] on theToeplitz algebras of quasi-lattice ordered monoids. Generalizing some of their results,we establish sufficient conditions for a KMS-state on NT (X) to be gauge-invariant,which can sometimes be applied to states at the critical inverse temperature and evenbelow it.

In Sect. 9 we consider a particular class of quasi-lattice ordered monoids, the right-angled Artin monoids. We show that for them the system of inequalities from Sect. 2reduces to a much smaller system, see Theorem 9.1. We leave open the problem to whatextent this result is true for other monoids. A solution to this problem would provide abetter answer to question (1).

Section 10 is motivated by the recent work of Kakariadis [17], which appeared whilewe were writing up this paper. In this work Kakariadis studies questions similar to (1)and (2) for finite rank product systems over Z

n+ for n ≥ 1. His approach is quite different

from ours and relies on a more refined decomposition of KMS-states than just intofinite/infinite types. Namely, a key idea in [17], partly inspired by [7], is that a state canbe of finite type with respect to one coordinate of Z

n+ and of infinite type with respect

to another. We explain how this idea works for product-monoids in our approach and asan application show how results in [17] and our results for Z

n+ can be quickly deduced

from each other, see Example 10.6.We conclude in Sect. 11 by considering some examples that have appeared in the

literature.Finally we would like to add that most results of the paper admit relatively straight-

forward generalizations to arbitrary quasi-free dynamics onNT (X) (see the discussionat the end of Sect. 1.4 for a more precise statement), but then the description of KMS-states on NT (X) is in terms of KMS-states on A instead of traces. There are several


reasons why we have chosen to work mostly with the gauge-type dynamics defined bya homomorphism N : P → (0,+∞) and confine ourselves to a few remarks on thegeneral case. The main one is that the gauge-type dynamics form the main focus ofthe current research. The general case would also require a few more prerequisites, inparticular some familiarity with modular theory and KMS-weights, which together withthe length of the paper might discourage the potential reader.

1. Preliminaries

1.1. Product systems and associated algebras. For a more thorough introduction intothese topics see [11].

Assume G is a group, P ⊂ G is a monoid generating G such that P ∩ P−1 = {e}.From time to time we will have to assume that P is finitely generated. By a generatingset of P we mean a set S ⊂ P such that every element of P\{e} can be written as aproduct of elements of S.

Define a partial order on G by

g ≤ h iff g−1h ∈ P.

Then G is said to be quasi-lattice ordered if any two elements g, h ∈ G with a commonupper bound have a least upper bound g ∨ h, see [25]. If g, h do not have a commonupper bound, we write g ∨ h = ∞. Therefore

gP ∩ hP ={

(g ∨ h)P, if g ∨ h < ∞,

∅, if g ∨ h = ∞.

For a finite subset J of P we let

qJ =∨

p∈J

p ∈ P ∪ {∞}. (1.1)

We say that a subset F of P is∨-closed if for any two elements p, q ∈ F with a commonupper bound in P we have p ∨ q ∈ F .

Let A be a C∗-algebra and X = (X p)p∈P be a collection of essential C∗-correspondences over A. Therefore each X p is a right C∗-Hilbert A-module equippedwith a left action of A by adjointable operators such that AX p = X p. The collection Xis called a product system over P if

(a) ∪p∈P X p has a semigroup structure such that for any ξ ∈ X p and ζ ∈ Xq wehave ξζ ∈ X pq , and the map ξ ⊗ ζ �→ ξζ extends to an isometric isomorphismX p ⊗A Xq ∼= X pq for all p, q ∈ P;

(b) Xe = A as a C∗-correspondence, and the products Xe×X p → X p, X p×Xe → X pcoincide with the structure maps of the A-bimodules X p.

Remark 1.1. In the literature a more general notion of product systems is often usedwhich does not require the C∗-correspondences X p to be essential, see, e.g., [5,30]. Forsome of our results, however, this assumption will be important. In the end it is notvery restrictive as we can always pass to the unitization A∼ of A and consider everyC∗-correspondence over A as an essential correspondence over A∼.


For p ≤ q we have a natural homomorphism ιqp : L(X p) → L(Xq) obtained by

identifying Xq with X p ⊗A X p−1q and mapping S ∈ L(X p) into S⊗1. Then X is calledcompactly aligned if for all p, q ∈ P with p ∨ q < ∞ we have

ιp∨qp (K(X p))ι

p∨qq (K(Xq)) ⊂ K(X p∨q).

By [11, Proposition 5.8], this assumption is satisfied if the left actions of A on X p areby generalized compact operators.

Assume now that B is a C∗-algebra and we are given linear maps ψp : X p → B. It isoften convenient to view such a collection (ψp)p∈P as one map ψ : X → B. It is calleda Toeplitz representation if

ψpq(ξζ ) = ψp(ξ)ψq(ζ ) for all ξ ∈ X p and ζ ∈ Xq

and

ψe(〈ξ, ζ 〉) = ψp(ξ)∗ψp(ζ ) for all ξ, ζ ∈ X p.

This in particular implies that ψe : A = Xe → B is a ∗-homomorphism.Given a Toeplitz representation ψ of X into B, we can define ∗-homomorphisms

ψ(p) : K(X p) → B by ψ(p)(θξ,ζ ) = ψp(ξ)ψp(ζ )∗, (1.2)

where θξ,ζ ∈ K(X p) is defined by θξ,ζ η = ξ 〈ζ, η〉.The main example of a Toeplitz representation is constructed as follows. Consider

the Fock module

F(X) =⊕

p∈P

X p

and define a representation � : X → L(F(X)) by

�(ξ)ζ = ξζ for ξ ∈ X p, ζ ∈ Xq .

We define the reduced Toeplitz algebra of X as the C∗-algebra T r (X) generated by theimage of � in L(F(X)).

Lemma 1.2. For any product system X, the C∗-algebra T r (X) is strictly dense inL(F(X)) = M(K(F(X))).

Proof. This is a standard argument. We first observe that if (ui )i is an approximate unitinK(X p), then the net (�(p)(ui ))i ⊂ T r (X) converges strictly to the projection f p ontothe submodule ⊕q∈pP Xq ⊂ F(X). Next, consider the finite subsets J ⊂ P\{e} andpartially order them by inclusion. Then the net

( ∏p∈J (1 − f p)

)J converges strictly

to the projection Qe onto Xe ⊂ F(X). Since for any ξ ∈ X p and ζ ∈ Xq we have�(ξ)Qe�(ζ )∗ = θξ,ζ , we can therefore conclude that the strict closure of T r (X) containsK(F(X)). This gives the result. ��

For any Toeplitz representation ψ : X → B we have the following useful property:if p ≤ q, ξ ∈ X p and ζ ∈ Xq , then

ψ(ξ)∗ψ(ζ ) = ψ(�(ξ)∗ζ ), (1.3)

which is easy to check on the elementary tensors ζ in Xq ∼= X p ⊗A X p−1q . Note that�(ξ)∗ζ ∈ X p−1q .


Any product system admits a universal Toeplitz representation X → T (X). In gen-eral, though, the Toeplitz algebra T (X) of X is too large to be interesting.

If X is compactly aligned, then a Toeplitz representation ψ : X → B is called Nicacovariant if

ψ(p)(S)ψ(q)(T ) ={

ψ(p∨q)(ιp∨qp (S)ι

p∨qq (T )), if p ∨ q < ∞,

0, otherwise.

In practice a more useful form of Nica covariance is often the following.

Lemma 1.3. A Toeplitz representation ψ : X → B of a compactly aligned productsystem is Nica covariant if and only if the following property holds: for ξ ∈ X p andζ ∈ Xq, we have

ψ(ξ)∗ψ(ζ ) = 0 if p ∨ q = ∞, (1.4)

ψ(ξ)∗ψ(ζ ) ∈ ψ(X p−1(p∨q))ψ(Xq−1(p∨q))∗ if p ∨ q < ∞. (1.5)

Proof. The forward implication is proved in [11, Proposition 5.10]. For the converse,assume (1.4) and (1.5) are satisfied and take S = θν,ξ ∈ K(X p) and T = θζ,η ∈ K(Xq).Then

ψ(p)(S)ψ(q)(T ) = ψ(ν)ψ(ξ)∗ψ(ζ )ψ(η)∗.

If p ∨ q = ∞, this expression is zero by (1.4).Assume now that p ∨ q < ∞. Since ψ(ξ)∗ψ(ζ ) ∈ ψ(X p−1(p∨q))ψ(Xq−1(p∨q))

∗by (1.5), if we take an approximate unit (ui )i in K(X p−1(p∨q)), we get the norm con-vergence

ψ(p−1(p∨q))(ui )ψ(ξ)∗ψ(ζ ) → ψ(ξ)∗ψ(ζ ).

We may assume that ui = ∑λ∈Ji θλ,λ for some finite sets Ji ⊂ X p−1(p∨q). Then

ψ(ν)ψ(p−1(p∨q))(ui )ψ(ξ)∗ =∑

λ∈Ji

ψ(ν)ψ(λ)ψ(λ)∗ψ(ξ)∗ = ψ(p∨q)(Si ),

where Si = ∑λ∈Ji θνλ,ξλ. It follows that

ψ(p)(S)ψ(q)(T ) = limi

ψ(ν)ψ(p−1(p∨q))(ui )ψ(ξ)∗ψ(ζ )ψ(η)∗

= limi

ψ(p∨q)(Si )ψ(ζ )ψ(η)∗.

Observe also that the net (Si )i in K(X p∨q) is bounded and converges to ιp∨qp (S) in the

strict topology on L(X p∨q) = M(K(X p∨q)).In a similar way we can take an approximate unit (u j ) j in K(Xq−1(p∨q)), insert

ψ(q−1(p∨q))(u j ) between ψ(ζ ) and ψ(η)∗, and get a bounded net (Tj ) j in K(X p∨q)such that

ψ(p)(S)ψ(q)(T ) = limi, j

ψ(p∨q)(Si )ψ(p∨q)(Tj ) = lim

i, jψ(p∨q)(Si Tj ),

with the convergence in norm, and ιp∨qq (T ) = lim j Tj in the strict topology onL(X p∨q).


To finish the proof it remains to show that

limi, j

ψ(p∨q)(Si Tj ) = ψ(p∨q)(ιp∨qp (S)ι

p∨qq (T )).

But this is true, since Si Tj → ιp∨qp (S)ι

p∨qq (T ) in the strict topology on K(X p∨q) and

hence ψ(p∨q)(Si Tj ) → ψ(p∨q)(ιp∨qp (S)ι

p∨qq (T )) in the strict topology on

ψ(p∨q)(K(X p∨q)). ��Remark 1.4. Condition (1.5) might not look like a typical relation, but in fact it dictateshow ψ(ξ)∗ψ(ζ ) can be approximated by elements of ψ(X p−1(p∨q))ψ(Xq−1(p∨q))

∗.Indeed, taking an approximate unit ui = ∑

λ∈Ji θλ,λ inK(X p−1(p∨q)) as in the previousproof, we get, using (1.3), that

ψ(ξ)∗ψ(ζ ) = limi

∑

λ∈Ji

ψ(λ)ψ(λ)∗ψ(ξ)∗ψ(ζ ) = limi

∑

λ∈Ji

ψ(λ)ψ(�(ζ )∗(ξλ))∗,

which gives the required approximation.

For every compactly aligned product system X there is a universal Nica covariantToeplitz representation iX : X → NT (X), see [11]. The C∗-algebra NT (X) is calledthe Nica-Toeplitz algebra of X ; it is denoted by Tcov(X) in op. cit.

For a ∈ A = Xe, we usually identify a with iX (a), so we view A as a C∗-subalgebraof NT (X). Properties (1.4) and (1.5) for ψ = iX imply that the space spanned by theelements iX (ξ)iX (ζ )∗ is dense in NT (X).

By [5, Proposition 3.5] the C∗-algebra NT (X) carries a full coaction

δ : NT (X) → NT (X) ⊗ C∗(G),

called the gauge coaction, such that

δ(iX (ξ)) = iX (ξ) ⊗ λp for ξ ∈ X p and p ∈ P.

For g ∈ G, denote by NT (X)g ⊂ NT (X) the subspace of elements of degree g withrespect to the coaction δ, that is, the set of elements S ∈ NT (X) such that δ(S) = S⊗λg .The elements iX (ξ)iX (ζ )∗, with ξ ∈ X p, ζ ∈ Xq and pq−1 = g, span a dense subspaceofNT (X)g . The fixed point subalgebraNT (X)e ⊂ NT (X) is also denoted by F andcalled the core subalgebra of NT (X). We have a unique gauge-invariant conditionalexpectation

E = (ι ⊗ τG) ◦ δ : NT (X) → F ,

where τG is the canonical trace on C∗(G).

Lemma 1.5. For any product system X, there is a unique reduced coaction δr of G onT r (X) such that

δr (�(ξ)) = �(ξ) ⊗ λp for all ξ ∈ X p and p ∈ P.

In other words, δr is an injective ∗-homomorphism T r (X) → T r (X) ⊗ C∗r (G) such

that

(ι ⊗ �) ◦ δr = (δr ⊗ ι) ◦ δr

and the space (1 ⊗ C∗r (G))δr (T r (X)) is dense in T r (X) ⊗ C∗

r (G).


Here � is the standard coproduct on C∗r (G) given by �(λg) = λg ⊗ λg . We call δr

again the gauge coaction.

Proof. Consider the right C∗-Hilbert A-module

F(X) ⊗ �2(G) =⊕

p∈P

X p ⊗ �2(G)

and the unitary operator V = ⊕p∈P 1X p ⊗ λp on it. For S ∈ L(F(X)) define

δr (S) = V (S ⊗ 1)V ∗ ∈ L(F(X) ⊗ �2(G)).

Ifwe viewL(F(X))⊗C∗r (G) as aC∗-subalgebra ofL(F(X)⊗�2(G)), then by definition

we immediately get that δr (�(ξ)) = �(ξ)⊗λp for ξ ∈ X p. Therefore the restriction of δrto T r (X) gives the required injective homomorphism T r (X) → T r (X) ⊗C∗

r (G) suchthat (ι⊗�)◦δr = (δr ⊗ι)◦δr . The density of (1⊗C∗

r (G))δr (T r (X)) in T r (X)⊗C∗r (G)

is clear, since products of �(ξ) and �(ξ)∗ for ξ ∈ X p, p ∈ P , span a dense subspace ofT r (X) and δr maps every such product into the same product tensored with λg for someg ∈ G. The uniqueness of δr is also obvious. ��

The fixed point subalgebra T r (X)e ⊂ T r (X) under the gauge coaction will also bedenoted by Fr and called the core subalgebra of T r (X). We have a faithful conditionalexpectation Er = (ι ⊗ τG) ◦ δr : T r (X) → Fr , cf. [19, Proposition 5.4].

If X is compactly aligned, then the Toeplitz representation � : X → L(F(X)) isNica covariant, hence we have a surjective ∗-homomorphism � : NT (X) → T r (X)

mapping iX (ξ) into �(ξ). Amenability criteria ensuring that � is injective are discussedin [11, Sections 7-8]. In particular, by [11, Corollary 8.2], � is injective if (G, P) is thefree product of quasi-lattice ordered groups (Gi , Pi ), i ∈ I , where every group Gi isamenable.

Independently of amenability of X we have the following property, which is a par-ticular case of [19, Corollary 6.4].

Lemma 1.6. For any compactly aligned product system X, the map � : NT (X) →T r (X) is injective on the core subalgebra F , so it defines an isomorphism F ∼= Fr .

Proof. For every finite subset F ⊂ P , denote by BF the sum of the subalgebrasi (p)X (K(X p)) ⊂ NT (X), p ∈ F . By [5, Lemma 3.6], if F is ∨-closed, then BF isa C∗-algebra. Since F is the inductive limit of these C∗-algebras, it is therefore enoughto show that �|BF is injective for every finite ∨-closed subset F ⊂ P . Take such a setF and a nonzero element S ∈ BF . By removing some elements of F if necessary, wemay assume that there exists a minimal element p in F such that the p-component ofS is nonzero. Let i (p)X (S′) be this component. Now let us look at the action of �(S) on

X p ⊂ F(X). Since � ◦ i (q)X = �(q) and we have

�(p)(S′)|X p = S′ and �(q)(S′′)|X p = 0

for any q �≤ p and S′′ ∈ K(Xq), we conclude that �(S)|X p = S′ �= 0. ��Corollary 1.7. Any gauge-invariant state φ on NT (X) factors through T r (X).


Proof. Letψ be the unique state onFr such thatψ ◦� = φ onF . By composing it withthe gauge-invariant conditional expectation Er : T r (X) → Fr , we can extend ψ to agauge-invariant state on T r (X), which we continue to denote byψ . Then the stateψ ◦�

vanishes on NT (X)g for all g �= e, hence it is gauge-invariant, and since it coincideswith φ on F , we conclude that ψ ◦ � = φ. ��

There are several otherC∗-algebras naturally associatedwith X .Wewill only considerthe following one, closely related to the Cuntz-Pimsner algebraOX defined in [11]. Fora compactly aligned product system X , denote by NO(X) the quotient of NT (X) bythe ideal generated by the elements a− i (p)X (ϕp(a)) for all p ∈ P and a ∈ ϕ−1

p (K(X p)),where ϕp : A → L(X p) is the homomorphism defining the left A-module structure onX p.

Remark 1.8. If P is directed (that is, p ∨ q < ∞ for all p, q ∈ P) and the maps ϕpare injective, then the Cuntz-Nica-Pimsner algebra NOX defined in [30] is a quotientof NO(X) (see, e.g., the computation in [30, Proposition 5.1(1)] together with [11,Proposition 5.4]). If, moreover, ϕp(A) ⊂ K(X p) for all p, then by [11, Proposition 5.4]

and [30, Corollary 5.2] the C∗-algebra NO(X) coincides with NOX , as well as withthe Cuntz-Pimsner algebra OX from [11].

The gauge coaction δ : NT (X) → NT (X) ⊗ C∗(G) descends to a full coactionof G on NO(X), which we continue to call the gauge coaction.

1.2. KMS-states. If σ = (σt )t∈R is a time evolution on a C∗-algebra A, recall that anelement a ∈ A is called analytic provided that the A-valued function t �→ σt (a) extendsto an analytic function onC. We letAa denote the subset of analytic elements, and recallthat it is a dense ∗-subalgebra of A, see, for example, [26, §8.12.1].

A state φ on A is called a σ -KMSβ -state if the equality

φ(ab) = φ(bσiβ(a)) (1.6)

holds for all a and b in a dense subset of analytic elements. It will be useful to have thefollowing simple lemma.

Lemma 1.9. Letσ be a time evolution on aC∗-algebra A. Suppose thatφ is aσ -invariantstate on A such that (1.6) is satisfied for all a in a set of analytic elements generating Aas a C∗-algebra and all b spanning a dense subspace of A. Then φ is a σ -KMSβ -state.

Proof. Let C be the set of all elements a ∈ Aa such that (1.6) holds for all b spanning adense subspace of A (depending on a). By linearity and continuity it follows that (1.6)holds for all a ∈ C and all b ∈ A. By assumption, C generates A as a C∗-algebra.

Taking adjoint in (1.6) gives

φ(b∗a∗) = φ(σiβ(a)∗b∗) = φ(σ−iβ(a∗)b∗)

whenever a ∈ C and b ∈ A. Since the adjoint of an analytic element is again in Aa, itfollows that b∗a∗ is analytic for every a ∈ C and b ∈ Aa. The assumed σt -invariancefor t ∈ R implies σz-invariance on analytic elements for all z ∈ C. Therefore

φ(σiβ(b∗)σiβ(a∗)) = φ(a∗σiβ(b∗)).


Renaming σiβ(b∗) as b, this shows that φ(a∗b) = φ(bσiβ(a∗)) for all a ∈ C and b ∈ Aa.Thus, C is closed under taking adjoint.

The set C is also closed under multiplication. Indeed, let a, c ∈ C. For each b ∈ Aawe have

φ(acb) = φ(cbσiβ(a)) = φ(bσiβ(a)σiβ(c)) = φ(bσiβ(ac)).

We have established that C is closed under adjoint and multiplication, so it is itself densein A. Therefore (1.6) holds for a dense set of analytic elements a and for all b ∈ A.Hence φ is a KMS-state. ��Assume now that (G, P) is a quasi-lattice ordered group and X is a compactly alignedproduct system over P . A homomorphism N : P → (0,+∞), with (0,+∞) viewed as agroup under multiplication, gives rise to a time evolution σ on the Nica-Toeplitz algebraNT (X) that satisfies

σt (iX (ξ)) = N (p)i t iX (ξ) for ξ ∈ X p, p ∈ P. (1.7)

It descends to time evolutions on NO(X) and T r (X). This is clear for NO(X), and inorder to show this for T r (X) consider the (possibly unbounded) operator DN on F(X)

such that

DN ξ = N (p)ξ for ξ ∈ X p. (1.8)

Then Ad DitN ◦ � = � ◦ σ , where (Ad Dit

N )(T ) = DitN T D−i t

N , so the required timeevolution on T r (X) is given by the restriction of Ad Dit

N . Note that it is well-defined

even when X is not compactly aligned and so the algebrasNT (X) and NO(X) are notdefined.

We will be interested in the KMS-states onNT (X) and NO(X) with respect to thedynamics defined by (1.7). The main problem is to understand the KMS-states on theformer algebra, the ones on the latter can then be obtained using the following lemma.

Lemma 1.10 (cf. [21, Theorem 2.5]). A KMS-state φ on NT (X) with respect to somedynamics onNT (X) factors through NO(X) if and only if φ(a) = φ(i (p)X (ϕp(a))) forall p ∈ P and a ∈ ϕ−1

p (K(X p)).

Proof. Clearly, the condition φ(a) = φ(i (p)X (ϕp(a))) is necessary forφ to factor through

NO(X). In order to see that it is sufficient, observe that i (p)X (ϕp(a)S) = a i (p)X (S) forall a ∈ A and S ∈ K(X p). It follows that if ϕp(a) ∈ K(X p), then

(a − i (p)X (ϕp(a)))∗(a − i (p)X (ϕp(a))) = a∗a − i (p)X (ϕp(a∗a)).

Hence φ vanishes on (a − i (p)X (ϕp(a)))∗(a − i (p)X (ϕp(a))). Since φ is a KMS-state, thecorresponding cyclic vector in the Hilbert space of the GNS-representation is separating,so it follows that a − i (p)X (ϕp(a)) is contained in the kernel of the GNS-representationfor all a ∈ ϕ−1

p (K(X p)). Therefore the GNS-representation of NT (X) factors through

NO(X) and hence φ factors through NO(X) as well. ��


Our main results will be about gauge-invariant KMS-states onNT (X). Note that byCorollary 1.7 a study of such states on NT (X) is equivalent to that on T r (X).

There is a much larger class of time evolutions on NT (X) than those defined byhomomorphisms N , which we will now briefly consider.

Assume A is equippedwith a timeevolutionγ = (γt )t∈R and everyC∗-correspondenceX p, p �= e, is equivariant, that is, we are given a strongly continuous one-parametergroup U (p) of isometries of the Banach space X p such that

〈U (p)t ξ,U (p)

t ζ 〉 = γt (〈ξ, ζ 〉) and U (p)t (aξ) = γt (a)U (p)

t ξ,

which implies that we also have U (p)t (ξa) = (U (p)ξ )γt (a). If we let U (e) = γ and

assume that the collection U = (U (p))p∈P is multiplicative in the sense that

U (pq)t (ξζ ) = (U (p)

t ξ)(U (q)t ζ ),

then we can define a quasi-free dynamics σU on NT (X) by

σUt (iX (ξ)) = iX (U (p)

t ξ)

for ξ ∈ X p, p ∈ P and t ∈ R.

1.3. Induction of traces and KMS-weights. If A is a C∗-algebra, τ is a tracial positivelinear functional on A, and Y is a right C∗-Hilbert A-module, then by [21, Theorem 1.1]we get a unique strictly lower semicontinuous, in general infinite, trace TrYτ on L(Y ),which we often denote simply by Trτ , such that

Trτ (θξ,ξ ) = τ(〈ξ, ξ 〉) for all ξ ∈ Y.

Explicitly, Trτ can be defined as follows. Assume (ui )i∈I is an approximate unit inK(Y )

such that, for every i , ui = ∑ξ∈Ji θξ,ξ for some finite set Ji ⊂ Y . Then

Trτ (T ) = supi

∑

ξ∈Ji

τ(〈ξ, T ξ 〉) = limi

∑

ξ∈Ji

τ(〈ξ, T ξ 〉) for T ∈ L(Y )+. (1.9)

In particular, if Y admits a (possibly infinite) Parseval frame (ξ j ) j∈J , meaning that

ξ =∑

j

ξ j 〈ξ j , ξ 〉 for all ξ ∈ Y,

then

Trτ (T ) =∑

j

τ(〈ξ j , T ξ j 〉) for T ∈ L(Y )+. (1.10)

Continuity of Trτ as a function of τ is a delicate issue, since Trτ is an infinite trace ingeneral, but the following simple observation will be enough for our purposes.

Lemma 1.11 (cf. [21, Proposition 2.4]). Assume A is a C∗-algebra, Y is a right C∗-Hilbert A-module and τ is a positive trace on A. Then


(i) for every T ∈ L(Y )+, the function ω �→ Trω(T ) is weakly∗ lower semicontinuous;in particular, if (τi )i is an increasing net of positive traces on A converging weakly∗to τ , then Trτi (T ) ↗ Trτ (T );

(ii) if Trτ (T ) < ∞ for some T ∈ L(Y )+, then the function ω �→ Trω(T ) is weakly∗continuous on the set of positive traces ω on A such that ω ≤ τ .

Proof. Part (i) is clear as the functions ω �→ ω(〈ξ, T ξ 〉) are weakly∗ continuous andthe function ω �→ Trω(T ) is the pointwise supremum of finite sums of such functions.

For part (ii), fix ε > 0 and take an operator S = ∑ξ∈J θξ,ξ ≤ 1, with J ⊂ Y finite,

such that Trτ (ST ) = ∑ξ∈J τ(〈ξ, T ξ 〉) > Trτ (T )−ε. Then for any positive traceω ≤ τ

and T ∈ L(Y )+ we have

Trω(T ) = Trω(ST ) + Trω((1 − S)T ) and 0 ≤ Trω((1 − S)T ) ≤ Trτ ((1 − S)T ) < ε.

It follows that if ω′ is another positive trace such that ω′ ≤ τ , then

|Trω(T ) − Trω′(T )| < |Trω(ST ) − Trω′(ST )| + 2ε

≤∑

ξ∈J

|ω(〈ξ, T ξ 〉) − ω′(〈ξ, T ξ 〉)| + 2ε.

When ω′ tends to ω in the weak∗ topology, then the last quantity converges to 2ε, andas ε > 0 was arbitrary, we conclude that Trω′(T ) → Trω(T ). ��

The following “induction in stages” type property of the traces Trτ is very useful,see [21, Proposition 1.2]. Assume Y is a right C∗-Hilbert A-module and Z is a C∗-correspondence over A such that the trace τ Z = TrZτ ◦ϕ on A is finite, where ϕ : A →L(Z) defines the left action of A. Then

TrYτ Z (T ) = TrY⊗AZ

τ (T ⊗ 1) for T ∈ L(Y )+. (1.11)

A few times it will be useful to remember that the construction of Trτ is part of amore general procedure of induction of KMS-weights [21]. Namely, assume that A isequipped with a time evolution γ and a right C∗-Hilbert A-module Y is equipped witha strongly continuous one-parameter group U of isometries such that

〈Utξ,Utζ 〉 = γt (〈ξ, ζ 〉).Define a time evolution σU on K(Y ) by σU

t (T ) = UtTU−t . Then for every γ -KMSβ -weight φ on 〈Y,Y 〉 there is a unique σU -KMSβ -weight � on K(Y ) such that

�(θξ,ξ ) = φ(〈Uiβ/2ξ,Uiβ/2ξ 〉)for all ξ in the domain of definition of Uiβ/2. This weight extends uniquely to a strictlylower semicontinuous weight IndUY φ onL(Y ); this extension was denoted by κφ in [21].Although the dynamics T �→ UtTU−t on L(Y ) is only strictly continuous, the weightIndUY φ satisfies the KMSβ condition with respect to it. When γ and U are trivial and φ

is a finite trace, then IndUY φ is exactly the trace TrYφ .By [21, Theorem 3.2], for every β ∈ R, the map φ �→ � defines a one-to-

correspondence between the γ -KMSβ -weights on 〈Y,Y 〉 and the σU -KMSβ -weightson K(Y ). A particular (but essentially equivalent) case of this correspondence is theclaim that if B is a C∗-algebra with time evolution σ and p ∈ M(B) is a σ -invariantfull projection, then the map φ �→ φ|pBp defines a one-to-one correspondence betweenthe σ -KMSβ -weights on B and those on pBp. We will only need injectivity of thismap, which is a rather simple consequence of the KMS-condition written in the formφ(aa∗) = φ(σiβ/2(a)∗σiβ/2(a)).


1.4. Generalized dual Ruelle transfer operators. Given a C∗-correspondence Y overa C∗-algebra A, we consider the operator FY mapping a positive trace τ on A into apositive, in general infinite, trace FY τ defined by

(FY τ)(a) = TrYτ (a) for a ∈ A+. (1.12)

Here, and in many other places, we omit the map ϕ : A → L(Y ) defining the left actionof A. To be pedantic, we should have written (FY τ)(a) = TrYτ (ϕ(a)).

Similarly to [21], the maps FY will play an important role in our study of KMS-states.As the following example shows, they can be considered as generalizations of the dualRuelle transfer operators.

Example 1.12. [9] Assume Z is a compact Hausdorff space and h : Z → Z is a sur-jective local homeomorphism. Then the Ruelle transfer operator L : C(Z) → C(Z)

(corresponding to the zero Hamiltonian) is defined by

(L f )(z) =∑

w∈h−1(z)

f (w).

Define a C∗-correspondence Y over A = C(Z) as follows. As a space we let Y =C(Z). The bimodule structure and the A-valued inner product are defined by

(a · ξ · b)(z) = a(z)ξ(z)b(h(z)), 〈ξ, ζ 〉(z) =∑

w∈h−1(z)

ξ(w)ζ(w)

for all a, b, ξ, ζ ∈ C(Z) and z ∈ Z . We claim that FY = L∗.Indeed, let (ρ j )

dj=1 be a partition of unity such that h is injective on each supp ρ j , and

take ξ j = √ρ j . Then {ξ j } j is a Parseval frame for Y (see, for example, [1, Lemma 5.2])

and∑

j

〈ξ j , a · ξ j 〉 = L(a).

By (1.10) we then get

(FY τ)(a) =∑

j

τ(〈ξ j , a · ξ j 〉) = τ(L(a)),

which is what we claimed.

Given a product system X over P , with Xe = A, we will write Fp instead of FXp .

In a similar way we will write Tr pτ instead of TrX pτ . We thus have

(Fpτ)(a) = Tr pτ (a) for a ∈ A+. (1.13)

Property (1.11) implies that

if Fqτ is finite, then FpFqτ = Fpqτ. (1.14)

We can nowmake a more precise statement on the possibility of extending the resultsof this paper from the time evolution on NT (X) defined by a homomorphism N togeneral quasi-free dynamics. The tracial states τ on A should be replaced by the γ -KMSβ -states ψ on A, and the expressions N (p)−β Tr pτ and N (p)−βFpτ should be


replaced by IndU(p)

X pψ and (IndU

(p)

X pψ)|A, respectively. Then the results involving no

conditions on N generalize withoutmuch effort to arbitrary quasi-free dynamics; wewillonlymake a few remarks throughout the paper indicating the necessaryminimal changes.On the other hand, with the results involving assumptions N (p) > 1 or N (p) ≥ 1 (and inSects. 2–10 these are onlyCorollary 7.5, Proposition 7.7,Corollary 7.8 andCorollary 9.5)one should be more careful, and we leave it to the interested reader or future work toclarify to what extent they can be generalized.

2. Restricting KMS-States to the Coefficient Algebra

Aiming to describe KMS-states φ on NT (X) in terms of their restrictions τ = φ|A toA = Xe, our first goal is to show that such states τ satisfy a number of inequalities.

Theorem 2.1. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyaligned product system of C∗-correspondences over P, with Xe = A. Consider thedynamics (1.7) on the Nica-Toeplitz algebraNT (X) of X defined by a homomorphismN : P → (0,+∞), and assume φ is a σ -KMSβ -state onNT (X) for some β ∈ R. Thenτ = φ|A is a tracial state on A satisfying the following condition:

τ(a) +∑

∅�=K⊂J

(−1)|K |N (qK )−β TrqKτ (a) ≥ 0 for all finite J ⊂ P\{e} and a ∈ A+,

(2.1)

where qK is given by (1.1) and the convention is that the summands corresponding toqK = ∞ are zero.

To be clear, our convention for the inclusion sign is that condition K ⊂ J allowsK = J . When we want to exclude the case K = J , we will write K � J .

Note that for J = {p} condition (2.1) gives N (p)−β Tr pτ (a) ≤ τ(a). Therefore partof this condition is that the summands in (2.1) are all finite and therefore there is noissue in having summands of different signs.

Before we turn to the proof of the theorem, let us recall a construction from [11].Consider the characteristic functions χpP of the sets pP ⊂ P . By the quasi-latticeassumption the set of such functions is closed under product, so the norm closure of thelinear span of χpP , p ∈ P , in �∞(P) is a C∗-algebra BP . Denote by BP the algebraof subsets of P generated by the sets pP , p ∈ P . Then the set of projections in BP isprecisely the set of characteristic functions of the elements of BP .

Given a representation π : NT (X) → B(H), for every p ∈ P , denote by f p theprojection onto the closed linear span of the images of π(iX (ξ)) for all ξ ∈ X p. Clearly,f p ∈ π(A)′. By [11, Propositions 4.1, 5.6 and 6.1], we have also

f p fq = f p∨q ,

with the convention f∞ = 0, so we get a representation Lπ of BP on H such thatLπ (χpP ) = f p.

Proof of Theorem 2.1. Since the dynamics is trivial on A, the positive linear functionalτ = φ|A is tracial. Since the C∗-correspondences X p are essential by our standingassumptions on product systems, an approximate unit in A is an approximate unit inNT (X), which implies that τ , being the restriction of a state, is a state itself.


Next, consider the GNS-triple (Hφ, πφ, vφ). By the above discussion we get a rep-resentation L = Lπφ : BP → B(Hφ) with image in πφ(A)′. Hence, for every a ∈ A+,we get a positive (finitely additive) measure μa on (P,BP ) defined by

μa(�) = (πφ(a)L(χ�)vφ, vφ) for � ∈ BP . (2.2)

We claim that μa(�), for � = ∩p∈J (P\pP), equals the expression on the left in (2.1),so condition (2.1) is simply a consequence of positivity of μa . Since

χ� =∏

p∈J

(1 − χpP ) = 1 +∑

∅�=K⊂J

(−1)|K |χqK P ,

our claim is equivalent to

μa(pP) = N (p)−β Tr pτ (a) for all a ∈ A+ and p ∈ P. (2.3)

Observe that by thedefinitionof f p = L(χpP ),wehave f pHφ = πφ(i (p)X (K(X p)))Hφ

(where i (p)X is given by (1.2) for ψ = iX : X → NT (X)). It follows that if (ui )i is an

approximate unit inK(X p), then the operators πφ(i (p)X (ui )) converge to f p in the strongoperator topology. Hence

μa(pP) = limi

φ(a i (p)X (ui )).

Now, by the KMS-condition, for any ξ, ζ ∈ X p, we have

φ(a i (p)X (θξ,ζ )) = φ(a iX (ξ)iX (ζ )∗)= N (p)−βφ(iX (ζ )∗a iX (ξ)) = N (p)−βτ(〈ζ, aξ 〉). (2.4)

So if we choose (ui )i such that ui = ∑ξ∈Ji θξ,ξ for some finite sets Ji ⊂ X , then

μa(pP) = limi

N (p)−β∑

ξ∈Ji

τ(〈ξ, aξ 〉).

This gives (2.3) by the definition of Tr pτ . ��Remark 2.2. In general a KMS-state φ on NT (X) is not uniquely determined by thetrace τ = φ|A. But at least the restriction of φ to the core subalgebra F ⊂ NT (X)

is completely determined by τ . Indeed, the elements of the form iX (ξ)iX (ζ )∗, withξ, ζ ∈ X p and p ∈ P , span a dense subspace of F , and on such elements the KMS-condition gives, as we already showed in (2.4), that

φ(iX (ξ)iX (ζ )∗) = N (p)−βτ(〈ζ, ξ 〉).Remark 2.3. An analogous result for general quasi-free dynamics is proved in an es-sentially identical way, but a computation similar to (2.4) goes smoother if one ob-serves that the projections f p lie in the centralizer of the normal state φ on M =πφ(NT (X))′′ defined by vφ and therefore the measure μa can be written as μa(�) =(πφ(a)J L(χ�)vφ, vφ), where J is the modular involution defined by φ. The reason isthat for general KMS-states the bilinear form φ(xσiβ/2(y)) = (πφ(x)Jπφ(y∗)vφ, vφ)

has better positivity properties than φ(xy).


Notice that formally condition (2.1) does not fully exploit positivity of the measures μaintroduced in the proof of Theorem 2.1, since we used only particular sets in BP insteadof all of them. The reason we formulated this condition as it stands, is that the additionalpositivity conditions we could get are already a consequence of (2.1). This is a byproductof Theorem 5.1 below, but let us show this directly. The following considerations willbe useful in several subsequent sections.

Recall thatwe defined generalized dualRuelle transfer operators Fp by (1.13).Denoteby T (A) ⊂ A∗ the closed subspace of bounded linear functionals φ such that φ(ab) =φ(ba) for all a, b ∈ A. Equivalently, T (A) is the linear span of tracial states. Then, ifFpτ is a finite trace for all p, we can define a linear operator from the space of functionsspanned by the characteristic functions of the sets in BP into T (A) by mapping χpP intoN (p)−βFpτ . (Herewe use that the characteristic functionsχpP are linearly independent,which can be quickly seen as follows. Assume that h = ∑

p∈F cpχpP = 0 for somefinite set F ⊂ P and complex numbers cp, but the set K = {p | cp �= 0} is nonempty.Then by evaluating h at a minimal point p of the set K we get cp = 0, which is acontradiction.)

By restricting the above linear operator to the characteristic functions of the sets � ∈BP we get a finitely additive T (A)-valued measure μ on (P,BP ) uniquely determinedby the property μ(pP) = N (p)−βFpτ . When τ = φ|A for a σ -KMSβ -state φ onNT (X), this vector-valued measure is related to the measures μa defined in (2.2) viathe identity

μa(�) = μ(�)(a). (2.5)

Condition (2.1) means precisely that

μ(�J ) ≥ 0, where �J =⋂

p∈J

(P\pP),

for all finite subsets J ⊂ P\{e}, and we want to show that this implies that μ is positive.The key point is that property (1.14) immediately gives positivity of μ on the sets p�Jas well. Indeed, since

χp�J = χpP +∑

∅�=K⊂J

(−1)|K |χpqK P ,

using property (1.14) we get

μ(p�J ) = N (p)−βFpτ +∑

∅�=K⊂J

(−1)|K |N (pqK )−βFpqK τ = N (p)−βFpμ(�J ),

(2.6)

from which we see that μ(p�J ) ≥ 0. Therefore positivity of μ is a consequence of part(i) of the following lemma. Part (ii) will be useful later.

Lemma 2.4. For any quasi-lattice ordered group (G, P), we have:

(i) every set in BP decomposes into a finite disjoint union of the sets pP and p�J ,where p ∈ P, J ⊂ P\{e} is a finite nonempty set and �J = ∩q∈J (P\qP);

(ii) for every � ∈ BP and p ∈ �, we have p ∈ p�J ⊂ � for some J .


Proof. (i) For any � ∈ BP there is a finite set F ⊂ P such that χ� is contained inthe linear span of the functions χpP , p ∈ F . By enlarging F we may assume that Fis ∨-closed. Then the linear span of the functions χpP , p ∈ F , forms an algebra. Theminimal projections of this algebra are the characteristic functions of sets of the form

( ⋂

p∈EpP

)∩

( ⋂

q∈F\E(P\qP)

), (2.7)

where E ⊂ F . Since pP ∩ qP = (p ∨ q)P , such a set is either empty or it has therequired form. Indeed, this is clear if E = ∅ or E = F . In the remaining cases we mayassume that qE = ∨p∈E p < ∞, as otherwise we get the empty set. Then the set in (2.7)equals qE P , if qE ∨ q = ∞ for all q ∈ F\E , or qE�J , where J = {q−1

E (qE ∨ q) | q ∈F\E, qE ∨ q < ∞}.

(ii) By part (i), if � contains p, then either p ∈ qP ⊂ � or p ∈ q�J ′ ⊂ � for someq and J ′. In either case we then have q ≤ p. Then in the first case we have p ∈ pP ⊂ �,so p ∈ p�J ⊂ � for any J . In the second case, if p ∨ qr = ∞ for all r ∈ J ′, thenpP∩q�J ′ = pP , so again p�J ⊂ � for any J . Otherwisewe have pP∩q�J ′ = p�J ,where J = {p−1(p ∨ qr) | r ∈ J ′, p ∨ qr < ∞}, and therefore p ∈ p�J ⊂ �. ��

We finish the section with a discussion of an important class of KMS-states for whichTheorem 2.1 does not give any nontrivial information.

Assume X is a not necessarily compactly aligned product system over P and τ0 is apositive trace on A = Xe such that

∑p∈P N (p)−β Tr pτ0(1) = 1. Recall that on the Fock

module level the dynamics σ is implemented by the unitaries DitN , where DN is given

by (1.8). By our assumption we have TrF(X)τ0 (D−β

N ) = 1. Hence, following [21], we canconsider the state

TrF(X)τ0

(·D−βN )

on L(F(X)), which we call a generalized Gibbs state. The tracial property of TrF(X)τ0

implies that it satisfies the KMSβ condition with respect to the dynamics Ad DitN . To be

more precise, there are no issues with the domain of definition of TrF(X)τ0 (·D−β

N ) and the

KMS-property when the operator D−βN is bounded. In the general case it is more fruitful

to think of TrF(X)τ0 (·D−β

N ) as Ind(Dit

N )t

F(X)τ0, that is, as the lower semicontinuous extension

of the KMSβ -weight induced from τ0 (viewed as a KMSβ -state with respect to the trivialdynamics) using the Fock module F(X) and the one-parameter unitary group (Dit

N )t .

But even then we will use the more suggestive notation TrF(X)τ0 (·D−β

N ).Since σt = Ad Dit

N on T r (X), by restriction we get a σ -KMSβ -state on T r (X).If X is compactly aligned, then by composing with � : NT (X) → T r (X) we get aσ -KMSβ -state Tr

F(X)τ0 (�(·)D−β

N ) onNT (X). The restriction τ of either of these statesto A is given by

τ(a) =∑

p∈P

N (p)−β(Fpτ0)(a) for a ∈ A.

The conclusion is that any tracial state of the form τ = ∑p∈P N (p)−βFpτ0 extends to

a σ -KMSβ -state on T r (X) or NT (X). As we will see later in Corollary 6.10(ii), if Xis compactly aligned, then these extensions are unique.


It is immediate that tracial states of the form τ = ∑p∈P N (p)−βFpτ0 satisfy con-

dition (2.1). Indeed, by Lemma 1.11(i) and (1.14) we have

N (p)−βFpτ =∑

q∈pP

N (q)−βFqτ0,

and from this it follows that for the T (A)-valued measure μ defined by μ(pP) =N (p)−βFpτ we have

μ(�) =∑

p∈�

N (p)−βFpτ0 for all � ∈ BP , (2.8)

which is obviously positive.

3. KMS-States on Reduced Toeplitz Algebras: The Case of Free Abelian Monoids

In this section we consider the monoid P = Zn+ (Z+ = {0, 1, 2, . . . }) for n ≥ 1. Denote

by e1, . . . , en the standard generators of Zn+.

We will write Fi instead of Fei for the operators (1.13). Then by (1.11), for anyi1, . . . , ik ∈ {1, . . . , n}, we have

Fi1 . . . Fik τ = Fei1+···+eik τ (3.1)

whenever the trace Fi j Fi j+1 . . . Fik τ is finite for all j = 2, . . . , k. In particular, Fi Fjτ =Fj Fiτ whenever both sides are well-defined.

Theorem 3.1. Assume X is a product system of C∗-correspondences over Zn+, Xe = A.

Consider the dynamics (1.7) on the reduced Toeplitz algebra T r (X) of X defined bya homomorphism N : Z

n+ → (0,+∞). Assume β ∈ R and τ is a tracial state on A

satisfying the following condition:

∏

i∈J

(1 − N (ei )−βFi )τ ≥ 0 for all nonempty subsets J ⊂ {1, . . . , n}. (3.2)

Then there exists a gauge-invariant σ -KMSβ -state φ on T r (X) such that φ|A = τ .

Note that if i1, . . . , ik are all different, then

ei1 + · · · + eik = ei1 ∨ · · · ∨ eik .

Together with (3.1) this shows that condition (3.2) is nothing else than condition (2.1)applied only to the subsets of {e1, . . . , en}.

For n = 1 and the homomorphism N : Z+ → (0,+∞) defined by N (1) = e,condition (3.2) is exactly condition (*) discussed in the introduction. For n ≥ 2 andparticular product systems condition (3.2) has appeared in [15] and [1] as a necessarycondition for a trace to be extendable to a KMS-state. It is called the subinvariancerelation in these papers. This name does not seem to be suitable for the more generalcondition (2.1).


Proof of Theorem 3.1. The case n = 1 follows from [21, Theorem 2.1]. For n ≥ 2 wecan use similar arguments.

Replacing N by Nβ we may assume that β = 1. If we also assume that there existsa positive trace τ0 on A such that

τ =∑

p∈Zn+

N (p)−1Fpτ0 =∞∑

k1,...,kn=0

N (e1)−k1 . . . N (en)

−kn Fk11 . . . Fkn

n τ0,

then by the discussion at the end of the previous section we get the required KMS-stateby restricting the generalized Gibbs state TrF(X)

τ0 (·D−1N ) to T r (X).

For an arbitrary tracial state τ satisfying (3.2), take ε > 0 and consider the perturbeddynamics σε defined by the homomorphism Nε : Z

n+ → (0,∞) such that Nε(ei ) =

eεN (ei ) for i = 1, . . . , n. Consider the trace

τ ε0 =

n∏

i=1

(1 − Nε(ei )−1Fi )τ.

This trace is positive, since by writing 1 − Nε(ei )−1Fi as

(1 − N (ei )−1Fi ) + (1 − e−ε)N (ei )

−1Fi ,

we see that

τ ε0 =

∑

J⊂{1,...,n}(1 − e−ε)n−|J |( ∏

j∈{1,...,n}\JN (e j )

−1Fj

)(∏

i∈J

(1 − N (ei )−1Fi )

)τ,

which is positive by (3.2).We claim that

τ =∑

p∈Zn+

Nε(p)−1Fpτ

ε0 .

To see this, take m ≥ 1 and consider the partial sum over all p ∈ Zn+ with coordinates

not greater than m, which rewrites as

n∏

i=1

(m∑

k=1

Nε(ei )−k Fk

i

)

τ ε0 =

n∏

i=1

(1 − e−(m+1)εN (ei )−(m+1)Fm+1

i )τ.

As m → ∞ these partial sums converge in norm to τ , since N (ei )−1Fiτ ≤ τ ande−(m+1)ε → 0. This implies our claim.

By the first part of the proof we conclude that there exists a gauge-invariant σε-KMS1-state φε on T r (X) such that φε|A = τ . Then any weak∗ cluster point of thestates φε as ε → 0 can be taken as the required state φ.

Remark 3.2. The above proof shows that, in the setting of Theorem 3.1, if a tracial state τ

on A satisfies (3.2) and has the property

limp→∞ N (p)−β Tr pτ (1) = 0, (3.3)

then τ = ∑p∈Z

n+N (p)−βFpτ0 for the positive trace τ0 = ∏n

i=1(1 − N (ei )−βFi )τ .

Conversely, it is easy to see that if τ = ∑p∈Z

n+N (p)−βFpτ0 for a positive trace τ0,


then (3.3) holds and τ0 = ∏ni=1(1−N (ei )−βFi )τ . Therefore we have a simple criterion

of presentability of τ in the form∑

p∈Zn+N (p)−βFpτ0, cf. [21]. This criterion and the

abundance of homomorphismsZn+ → (0,∞), allowing us to easilymodify the dynamics

in order to enforce (3.3), are what make the case P = Zn+ special and the above proof

work. We will return to the study of traces of the form∑

p∈P N (p)−βFpτ0 for generalquasi-lattice ordered groups in Sect. 7.

If X is in addition compactly aligned, then NT (X) is well-defined and we get thefollowing result.

Corollary 3.3. Assume X is a compactly aligned product system of C∗-correspondencesover the monoid Z

n+, Xe = A, and consider the dynamics (1.7) onNT (X) defined by a

homomorphism N : Zn+ → (0,+∞). Then, for every β ∈ R, the map φ �→ φ|A defines

a one-to-one correspondence between the gauge-invariant σ -KMSβ -states on NT (X)

and the tracial states on A satisfying (3.2).

Proof. If φ is a gauge-invariant σ -KMSβ -state on NT (X), then by Theorem 2.1 itsrestriction τ to A is a tracial state satisfying (2.1). By the discussion right after the for-mulation of Theorem 3.1, condition (3.2) is formally weaker than (2.1). By the gauge-invariance, φ is determined by its restriction to the core subalgebra F , while by Re-mark 2.2 this restriction is completely determined by τ . Therefore themapφ �→ τ = φ|Ais an injective map from the set of gauge-invariant σ -KMSβ -states on NT (X) into theset of tracial states on A satisfying (3.2). By Theorem 3.1 this map is also surjective,since T r (X) is a quotient ofNT (X) (in fact, as we know, the two algebras coincide byamenability of Z

n).

As a byproduct we see that for P = Zn+ and compactly aligned product systems

conditions (2.1) and (3.2) are equivalent. This can be shown directly and without usingthe compact alignment assumption. We will return to this conclusion and prove a moregeneral result in Sect. 9.

In Example 11.4 we will show that in general the system of inequalities (3.2) cannotbe replaced by yet a smaller system.

4. States on Quasi-Lattice Graded Algebras

In order to prove an analogue of Theorem 3.1 for general quasi-lattice ordered groupswe will extend a trace τ satisfying (2.1) to the core subalgebra. The main problem willbe to show that such an extension is positive, and in this section we prove a result whichwill allow us to do that. It will be convenient to work in a more abstract setting than thatof Nica-Toeplitz algebras.

By a quasi-lattice we mean a partially ordered set I such that any two elementsp, q ∈ I either have a least upper bound, which we denote p ∨ q, or do not have acommon upper bound at all, in which case we write p ∨ q = ∞.

Definition 4.1. A quasi-lattice graded C∗-algebra is a C∗-algebra F together with adense ∗-subalgebra B = ⊕p∈I Bp, which we call the algebraic core of F , such that I isa quasi-lattice and the following conditions are satisfied:

(a) each space Bp, p ∈ I , is a C∗-subalgebra of F ;(b) for all p, q ∈ I , we have BpBq ⊂ Bp∨q , with the convention B∞ = 0;(c) for all q ≤ p, we have Bq Bp = Bp.


The followinguseful observation is a straightforwardgeneralizationof [5,Lemma3.6].

Lemma 4.2. IfF is a quasi-lattice gradedC∗-algebrawith algebraic coreB = ⊕p∈I Bp,then for every finite∨-closed subset J of I , the space BJ = ⊕p∈J Bp is a C∗-subalgebraof F .

Proof. The proof relies only on conditions (a) and (b) and uses induction on the size ofJ along the same lines as that of [5, Lemma 3.6]. (The key point is that the sum of aC∗-subalgebra and a closed ideal of a C∗-algebra is always closed.) ��

Assume now that for every p ∈ I we are given a positive linear functional φp onBp. Our goal is to show that an analogue of (2.1) provides a sufficient condition forpositivity of the linear functional φ = ⊕pφp on B. We need to introduce some notationto formulate the precise result.

Extend φp to a linear functional ψp on ⊕q Bq as follows. If q �≤ p, then we putψp|Bq = 0. On the other hand, if q ≤ p, then by assumption (b) we have a ∗-homomorphism Bq → M(Bp). Composing the canonical extension of φp to a positivelinear functional on M(Bp)with this homomorphism, we get a positive linear functionalon Bq . We take this functional as ψp|Bq .Lemma 4.3. For any choice of positive linear functionals φp on Bp, the linear function-als ψp are positive on B, that is, ψp(x∗x) ≥ 0 for all x ∈ B.Proof. Fix p ∈ I . First of all observe that since the closure of ⊕q≤pBq in F is a C∗-algebra and Bp is its ideal, the functional φp extends in a canonical way to a positivelinear functional on this C∗-algebra. On the dense ∗-subalgebra ⊕q≤pBq this positivelinear functional coincides with ψp by construction.

Next, observe that if y ∈ Bq for some q �≤ p, then q ∨ r �≤ p for any r ∈ I , andhence ψp(xy) = 0 for all x ∈ B.

Now, take any x ∈ ⊕q≤pBp and y ∈ ⊕q �≤pBq . Then by the above two observationswe have

ψp((x + y)∗(x + y)) = ψp(x∗x) ≥ 0.

Thus ψp is indeed positive on B. ��Proposition 4.4. Assume F is a quasi-lattice graded C∗-algebra with algebraic coreB = ⊕p∈I Bp. Assume that we are given positive linear functionals φp on Bp such that,for every p ∈ I and every finite subset J ⊂ {q|q > p}, we have

φp(b) +∑

∅�=K⊂J

(−1)|K |ψqK (b) ≥ 0 for all positive elements b ∈ Bp, (4.1)

where the functionalsψqK are defined as explained above, with the conventionψ∞ = 0.Then the linear functional φ = ⊕pφp on B is positive. Furthermore, if I has a smallestelement e, and φe is a state on Be, then φ extends to a state on F .

For the proof we need the following lemma.

Lemma 4.5. Assume J is a finite quasi-lattice and f is a complex-valued function on J .Define a new function f on J by

f (p) = f (p) +∑

∅�=K⊂{q∈J |q>p}(−1)|K | f (qK ),


with the convention f (∞) = 0. Then, for every p ∈ J , we have

f (p) =∑

q≥p

f (q).

Proof. For every p ∈ J , denote by Jp the set {q | q ≥ p}. The characteristic functionsχJp of these sets, being linearly independent, form a basis of the space of functions on J .It follows that there exists a unique complex measure μ on J such that μ(Jp) = f (p).For every p ∈ J , we have {p} = Jp\ ∪q>p Jq , so that

χ{p} =∏

q>p

(χJp − χJq ) = χJp +∑

∅�=K⊂{q∈J |q>p}(−1)|K |χJqK

.

From this we see that μ({p}) = f (p) for all p ∈ J . Then

f (p) = μ(Jp) =∑

q∈Jp

μ({q}) =∑

q∈Jp

f (q),

which is what we need. ��Proof of Proposition 4.4. By Lemma 4.2, it suffices to show that φ is positive on the∗-subalgebra

BJ = ⊕p∈J Bp ⊂ B

for all finite ∨-closed subsets J ⊂ I . With such a J fixed, for every p ∈ J define alinear functional ηp on BJ as follows. If q �≤ p, then we let ηp|Bq = 0, and if q ≤ p,we put

ηp(x) = ψp(x) +∑

∅�=K⊂{r∈J |r>p}(−1)|K |ψqK (x) for x ∈ Bq .

We claim that the functionals ηp are positive. Indeed, by our assumption (4.1), thelinear functional ηp|Bp is positive. By Lemma 4.3, applied to the set J and the func-tional ηp|Bp instead of I and φp, this functional extends to a positive linear functionalωp on BJ . Specifically, we have ωp|Bq = 0 for q �≤ p, and

ωp(x) = limi

ηp(xui )

for x ∈ Bq with q ≤ p, where {ui }i is an approximate unit in Bp. By assumption (c) inDefinition 4.1, for every r ≥ p, the images of ui in M(Br ) converge strictly to 1. Henceψr (x) = limi ψr (xui ) for all x ∈ Bq with q ≤ p. From this we conclude that ωp = ηp,hence ηp is indeed positive.

Finally, we claim that φ|BJ = ∑p∈J ηp, which clearly implies positivity of φ|BJ .

Indeed, take s ∈ J , x ∈ Bs and consider the function f on J defined by f (p) = ψp(x).Using the notation of Lemma 4.5, we have ηp(x) = f (p) for all p ≥ s. Hence, by thatlemma,

∑

p∈J : p≥s

ηp(x) = f (s) = ψs(x) = φs(x).


On the other hand, by definition, ηp(x) = 0 for p �≥ s. Therefore∑

p∈J ηp(x) = φs(x),and our claim is proved.

It remains to prove the last statement of the proposition. By Lemma 4.2, for everyfinite ∨-closed subset J of I containing e, the ∗-subalgebra BJ of F is closed. Since eis assumed to be the smallest element of I , by assumption (c) an approximate unit ofBe is an approximate unit of BJ . Since we already know that φ|BJ is a positive linearfunctional, it follows that

‖φ|BJ ‖ = ‖φ|Be‖ = ‖φe‖ = 1,

so φ|BJ is a state. Since F is the closure of the union of such C∗-subalgebras BJ , weconclude that φ extends by continuity to a state on F . ��

It might be useful also to observe the following, although in our applications this willbe subsumed by stronger statements.

Proposition 4.6. Assume F is a quasi-lattice graded C∗-algebra with algebraic coreB = ⊕p∈I Bp, andwe are given tracial positive linear functionals φp on the C∗-algebrasBp. Then the linear functional φ = ⊕pφp on B is tracial as well.

More generally, if we are given a one-parameter group (σt )t∈R of automorphisms ofF leaving the subalgebras Bp invariant and σ -KMSβ positive linear functionals φp onBp, then φ is σ -KMSβ , meaning that φ(ab) = φ(bσiβ(a)) for all σ -analytic elementsa, b ∈ B.

Proof. Take x ∈ Bq and y ∈ Br . We want to show that φ(xy) = φ(yx). If q ∨ r = ∞,then xy = yx = 0 and there is nothing to prove. So assume p = q ∨ r < ∞. Aswe already used in the proof of Lemma 4.3, Bp is an ideal of the C∗-algebra ⊕s≤p Bs .Hence φp extends in a canonical way to a tracial positive linear functional ψp on thisC∗-algebra. Then

φ(xy) = φp(xy) = ψp(xy) = ψp(yx) = φp(yx) = φ(yx).

The second part of the proposition is proved in a similar way. We just want to remarkthat since the subspaces BJ = ⊕p∈J Bp ⊂ F are closed for finite ∨-closed sets J byLemma 4.2, the topology on them coincides with the direct product topology, and thisimplies that an element of B is σ -analytic in F if and only if its Bp-components areσ -analytic for all p. ��

5. KMS-States on Nica-Toeplitz Algebras: The General Case

Using the results of the previous section we can now prove an analogue of Corollary 3.3for arbitrary quasi-lattice ordered groups.

Theorem 5.1. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyaligned product system of C∗-correspondences over P, with Xe = A. Consider thedynamics (1.7) on NT (X) defined by a homomorphism N : P → (0,+∞). Then, forevery β ∈ R, the map φ �→ φ|A defines a one-to-one correspondence between thegauge-invariant σ -KMSβ -states onNT (X) and the tracial states on A satisfying (2.1).

Proof. We only have to prove surjectivity of the map φ �→ φ|A, as injectivity followsfromRemark 2.2. Thus we have to show that every tracial state τ satisfying (2.1) extendsto a gauge-invariant σ -KMSβ -state φ on NT (X).


In order to define φ we first extend τ to the core subalgebra F and then use thegauge-invariant conditional expectationNT (X) → F to define φ on the whole algebraNT (X). By Remark 2.2 we know that the required extension on F must be given by

φ(iX (ξ)iX (ζ )∗) = N (p)−βτ(〈ζ, ξ 〉) for ξ, ζ ∈ X p. (5.1)

In order to see that such a state indeed exists we apply Proposition 4.4. For this letBp = i (p)X (K(X p)) ⊂ F for p ∈ P , where, as before, i (p)X denotes the canonical injectivehomomorphism K(X p) → NT (X), see (1.2). As the positive linear functionals φp wetake

φp(i(p)X (T )) = N (p)−β Tr pτ (T ) for T ∈ K(X p).

We have to check that condition (4.1) is satisfied for all p ∈ P and all finite J ⊂{q|q > p}. For p = e this is condition (2.1). For general p it suffices to establish (4.1)on all elements of the form b = iX (ξ)iX (ξ)∗ ∈ Bp, since finite sums of the elementsθξ,ξ are dense in K(X p)+. We have

φp(iX (ξ)iX (ξ)∗) +∑

∅�=K⊂J

(−1)|K |ψqK (iX (ξ)iX (ξ)∗)

= N (p)−β Tr pτ (θξ,ξ ) +∑

∅�=K⊂J

(−1)|K |N (qK )−β TrX p⊗AX p−1qKτ (θξ,ξ ⊗ 1).

By definition and property (1.11) of induced traces this equals

N (p)−βτ(〈ξ, ξ 〉) +∑

∅�=K⊂J

(−1)|K |N (qK )−β Tr p−1qK

τ (〈ξ, ξ 〉),

and this is positive by condition (2.1) applied to a = 〈ξ, ξ 〉 and the set p−1 J . Thus agauge-invariant state φ satisfying (5.1) indeed exists.

It remains to check the KMS-condition for φ. For this we apply Lemma 1.9 to theanalytic elements a = iX (ξ), ξ ∈ X p, generating NT (X) as a C∗-algebra. Taking b =iX (ζ )iX (η)∗, with ζ ∈ Xq and η ∈ Xr , we have to check that φ(ab) = N (p)−βφ(ba).Consider two cases.

Case 1: pq �= r . Then ab and ba are of degrees pqr−1 and qr−1 p, respectively, sothey are in the kernel of the gauge-invariant conditional expectation NT (X) → F andtherefore the equality φ(ab) = N (p)−βφ(ba) holds for trivial reasons.

Case 2: pq = r . Then

φ(ab) = φ(iX (ξζ )iX (η)∗) = N (pq)−βτ(〈η, ξζ 〉).Using (1.3) we also get

φ(ba) = φ(iX (ζ )iX (η)∗iX (ξ)) = φ(iX (ζ )iX (�(ξ)∗η)∗)= N (q)−βτ(〈�(ξ)∗η, ζ 〉) = N (q)−βτ(〈η, ξζ 〉),

so we have φ(ab) = N (p)−βφ(ba) in this case too. ��


Remark 5.2. The above theorem describes only gauge-invariant KMS-states. If one isinterested in the set of all KMS-states, it is still a good idea to start with the gauge-invariant ones. The reason is that if φ is a σ -KMSβ -state on NT (X) for some β ∈ R,then by Theorems 2.1 and 5.1 there exists a unique gauge-invariant σ -KMSβ -state ψ onNT (X) such thatψ = φ on A, moreover, we then haveψ = φ onF . The existence ofψdoes not actually require any of the above results, since it is not difficult to check directlythat the state ψ = φ ◦ E , where E = (ι ⊗ τG) ◦ δ : NT (X) → F is the gauge-invariantconditional expectation, satisfies the σ -KMSβ -condition.

Let us now consider the implications of Theorem 5.1 for the quotient NO(X) ofNT (X).

Corollary 5.3. In the setting of Theorem 5.1, for every β ∈ R, the map φ �→ φ|A definesa one-to-one correspondence between the gauge-invariant σ -KMSβ -states on NO(X)

and the tracial states on A such that (2.1) is satisfied and

N (p)−β Tr pτ (a) = τ(a) for all a ∈ ϕ−1p (K(X p)), p ∈ P,

where ϕp : A → L(X p) is the map defining the left A-module structure on X p.

Proof. If φ is a σ -KMSβ -state onNT (X) and τ = φ|A, then it follows from (2.4) that

N (p)−β Tr pτ (S) = φ(i (p)X (S)) for all S ∈ K(X p).

Therefore the condition N (p)−β Tr pτ (a) = τ(a) for a ∈ ϕ−1p (K(X p)) means precisely

that

φ(a − i (p)X (ϕp(a))

) = 0.

By Lemma 1.10 the latter condition is necessary and sufficient for φ to factor throughNO(X). The result now follows from Theorem 5.1. ��

Under extra assumptions this corollary takes a particularly simple form.

Corollary 5.4. In the setting of Theorem 5.1, assume in addition that P is directed andϕp(A) ⊂ K(X p) for all p ∈ P. Fix a generating set S ⊂ P\{e}. Then, for every β ∈ R,the map φ �→ φ|A defines a one-to-one correspondence between the gauge-invariantσ -KMSβ -states on NO(X) and the tracial states τ on A such that

N (p)−β Tr pτ (a) = τ(a) (5.2)

for all a ∈ A and p ∈ S.

Proof. By Corollary 5.3 we just have to check that if τ is a tracial state such that (5.2)holds for all a ∈ A and p ∈ S, then (5.2) holds for all p ∈ P and (2.1) is satisfied aswell.

The first assertion follows from (1.14). For the second one, take a finite subset J ⊂P\{e}. Then, using that qK < ∞ for all ∅ �= K ⊂ J by assumption, for every a ∈ Awe get

τ(a) +∑

∅�=K⊂J

(−1)|K |N (qK )−β TrqKτ (a) = τ(a)∑

K⊂J

(−1)|K | = 0,

so (2.1) is satisfied. ��


6. A Wold-Type Decomposition of KMS-States and Traces

In this section we will clarify and extend the decomposition of KMS-states of Pimsner-Toeplitz algebras into finite and infinite parts from [21] to Nica-Toeplitz algebras.

We start with the following surely known result.

Proposition 6.1. Assume (G, P) is a quasi-lattice ordered group, X is a compactlyalignedproduct systemofC∗-correspondences over P,with Xe = A, andπ : NT (X) →B(H) is a representationofNT (X). Consider the vonNeumannalgebra M = π(NT (X))′′.Then there exists a central projection z ∈ M such that zH is the space of the largestsubrepresentation of π induced from A by the Fock module F(X).

Proof. Let Q be the projection onto the space

{v ∈ H : π(iX (ξ))∗v = 0 for all ξ ∈ X p, p �= e} (6.1)

of “vacuum vectors”. The space QH is invariant under M ′, hence Q ∈ M . It is alsoclear that the space QH is invariant under π(A).

By virtue of (1.4) and (1.5) (for ψ = iX ), if ξ ∈ X p, ζ ∈ Xq and p �= q, then thespaces iX (ξ)QH and iX (ζ )QH are orthogonal. It follows that the map

U : F(X) ⊗A QH → H, ξ ⊗ v �→ π(iX (ξ))v,

is isometric. Its image is obviously invariant under the operators π(iX (ξ)), but usingagain (1.4) and (1.5) we see that it is invariant under the operators π(iX (ξ))∗ as well.Therefore the image of U coincides with the invariant subspace MQH ⊂ H , so therestriction of π to this subspace is unitarily equivalent to the representation induced bythe Fock module from the representation of A on QH .

Since Q ∈ M , the projection z onto MQH is a central projection in M . Finally, if asubrepresentation of π is induced by the Fock module, then its space of vacuum vectorsis contained in QH and hence the entire space of the subrepresentation is contained inzH . ��

The property of a representation of NT (X) to be induced by the Fock module canbe reformulated as a continuity property as follows.

Lemma 6.2. A representation π : NT (X) → B(H) is induced from a representationof A by the Fock module if and only if it factors through T r (X) and the representationof T r (X) we thus get is strict-strong continuous, that is, it is continuous with respect tothe strict topology on T r (X) ⊂ L(F(X)) and the strong operator topology on B(H).

Proof. The “only if” implication is obvious. To prove the “if” part, assume we havea representation π of T r (X) which is strict-strong continuous. Since T r (X) is strictlydense in L(F(X)) by Lemma 1.2, it extends to a strict-strong continuous representationof L(F(X)). The latter representation is completely determined by its restriction toK(F(X)). But since the right C∗-Hilbert A-module F(X) is full, any representation ofK(F(X)) is induced from a representation of A by the module F(X). ��Remark 6.3. This lemma makes it obvious that a subrepresentation of a representationof NT (X) induced by the Fock module is itself induced by the Fock module. Theproof shows more: the induction by F(X) defines an equivalence of the category ofrepresentations of A = Xe onto a full subcategory of the category of representations ofNT (X).


Definition 6.4. A positive linear functional φ onNT (X) is said to be of finite type if theassociated GNS-representation is induced by the Fock module. It is said to be of infinitetype if the associated GNS-representation has no nonzero subrepresentations inducedby the Fock module.

Similarly to Lemma 6.2, these notions can be formulated as continuity/discontinuityproperties. To state the precise result, recall that� denotes the canonicalmapNT (X) →T r (X).

Lemma 6.5. For any positive linear functional φ on NT (X) we have:

(i) φ is of finite type if and only if φ = ψ ◦ � for a strictly continuous positive linearfunctional ψ on T r (X);

(ii) φ is of infinite type if and only if there is no nonzero strictly continuous positivelinear functional ψ on T r (X) such that φ ≥ ψ ◦ �.

Proof. (i) Assume φ is of finite type. Consider the associated GNS-triple (Hφ, πφ, vφ).Since by assumption πφ is induced by the Fock module, we have πφ = π ◦ � for astrict-strong continuous representation π of T r (X). Then the positive linear functionalψ = (π(·)vφ, vφ) on T r (X) is strictly continuous and φ = ψ ◦ �.

Conversely, assume φ = ψ ◦ � for a strictly continuous positive linear functional ψon T r (X). Since T r (X) is strictly dense in L(F(X)), ψ extends to a strictly continuouspositive linear functional onL(F(X)), which we continue to denote byψ . Let (H, π, v)

be the GNS-triple associated with ψ |K(F(X)), and consider the unique extension of π toL(F(X)), which we also continue to denote by π . As we already used in the proof ofLemma 6.2, any representation of K(F(X)) is induced by the Fock module. It followsthat the representation π of L(F(X)), hence also π ◦�, is induced by the Fock module.At the same time we have ψ = (π(·)v, v) on L(F(X)), since this equality holds onK(F(X)) and both sides are strictly continuous. It follows that (H, π ◦ �, v) is theGNS-triple associated with ψ ◦ � = φ. Hence φ is of finite type.

(ii) Assume φ is of infinite type. If φ ≥ ψ ◦ � for some nonzero strictly continuouspositive linear functional ψ on T r (X), then ψ ◦ � is of finite type by part (i), so theassociated GNS-representation is induced by the Fock module. But this representationis a subrepresentation of the GNS-representation associated with φ. This contradicts theassumption that φ is of infinite type.

For the converse, assume φ is not of infinite type. Consider the associated GNS-triple(Hφ, πφ, vφ). By the assumption there exists a nonzero projection f ∈ πφ(NT (X))′such that the restriction on πφ to f Hφ is induced by the Fock module. In particular, thisrestriction factors through T r (X) and defines a strict-strong continuous representation π

of T r (X) on f Hφ . Then ψ = (π(·) f vφ, vφ) is a nonzero strictly continuous positivelinear functional on T r (X). As π ◦ � = πφ(·)| f Hφ , we have ψ ◦ � ≤ φ. ��

Note that Lemmas 6.2 and 6.5, as opposed to Proposition 6.1, are quite formal anddo not use any properties of NT (X) apart from strict density of T r (X) in L(F(X)).

Proposition 6.6. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyaligned product system of C∗-correspondences over P. Then any positive linear func-tional φ onNT (X) has a unique decomposition φ = φ f +φ∞ where φ f is of finite typeand φ∞ is of infinite type.

Proof. Let (Hφ, πφ, vφ) be the GNS-triple associated with φ. Consider the von Neu-mann algebra M = πφ(NT (X))′′ and let z ∈ M be the central projection given by


Proposition 6.1. Then letting φ f = (πφ(·)zvφ, vφ) and φ∞ = (πφ(·)(1 − z)vφ, vφ) weget the required decomposition of φ.

In order to prove that the decomposition is unique it suffices to establish the followingproperties of the functionals φ f and φ∞ defined above: if ψ ≤ φ is of finite type, thenψ ≤ φ f , and if ψ ≤ φ is of infinite type, then ψ ≤ φ∞.

So assume we have a positive linear functional ψ such that ψ ≤ φ. Let x ∈ M ′be the unique element such that 0 ≤ x ≤ 1 and ψ = (πφ(·)xvφ, vφ), and let f ∈ M ′be the support projection of x . Then as the GNS-triple associated with ψ we can take( f Hφ, πφ | f Hφ , x1/2vφ).

Now, if ψ is of finite type, then we must have f ≤ z. Hence x ≤ z and thereforeψ ≤ φ f . On the other hand, ifψ is of infinite type, then z f = 0, since the representationπφ |z f Hφ is induced by the Fock module and is contained in the GNS-representation ofψ , so it must be zero. Hence x ≤ 1 − z and therefore ψ ≤ φ∞. ��

The construction of φ f and φ∞ implies the following.

Corollary 6.7. If in the setting of Proposition 6.6 the positive linear functionalφ satisfiesthe σ -KMSβ condition for some time evolution σ on NT (X) and β ∈ R, then φ f andφ∞ also satisfy the σ -KMSβ condition.

Proof. Using the same notation as in the proof of Proposition 6.6, consider the normalpositive linear functional φ on M defined by the vector vφ . Then φ is faithful and its

modular group σ φ satisfies πφ ◦ σ−βt = σφt ◦ πφ . Since φ satisfies the σ φ-KMS−1

condition, it follows that the functionals φ f = φ(πφ(·)z) and φ∞ = φ(πφ(·)(1 − z))satisfy the σ -KMSβ condition. ��

Our next goal is to understand what the decomposition φ = φ f + φ∞ of a σ -KMSβ -state, where σ is given by (1.7), means at the level of the trace τ = φ|A.

Recall that we denote by Fp the operator mapping a trace τ on A into Tr pτ |A.Theorem 6.8. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyaligned product system of C∗-correspondences over P, with Xe = A. Consider thedynamics (1.7) on NT (X) defined by a homomorphism N : P → (0,+∞), and fixβ ∈ R. Assume φ is a σ -KMSβ -state on NT (X), consider the finite part φ f of φ, andlet φ f be the unique strictly continuous positive linear functional on L(F(X)) such thatφ f = φ f ◦ �. Consider the tracial state τ = φ|A and define a positive trace τ0 onM(A) ∼= L(Xe) by

τ0(a) = φ f (aQe),

where Qe ∈ L(F(X)) is the projection onto Xe ⊂ F(X). Then

(i) for every a ∈ M(A),

τ0(a) = limJ

(τ(a) +

∑

∅�=K⊂J

(−1)|K |N (qK )−β TrqKτ (a)), (6.2)

where the limit is over the net of finite subsets J (partially ordered by inclusion) ofany fixed generating set S ⊂ P\{e} and we consider the canonical extension of τ

to M(A);


(ii) the trace τ f = φ f |A is given by

τ f =∑

p∈P

N (p)−βFpτ0;

(iii) the functional φ f is given by

φ f = TrF(X)τ0

(�(·)D−βN ), (6.3)

where DN is the operator on F(X) defined by (1.8).

Proof. (i)Using thenotation from theproof ofProposition6.6,wehaveφ f = (πφ(·)zvφ, vφ).The representation πφ |zHφ factors through L(F(X)), and by the construction of theprojection z in the proof of Proposition 6.1, the strictly continuous representation ofL(F(X)) we thus get maps Qe into the projection Q onto the space (6.1) of vacuumvectors. It follows that

τ0(a) = φ f (aQe) = (πφ(a)Qvφ, vφ) for a ∈ A.

Next, recall from the proof of Theorem 2.1 that we also have a representationL : BP → B(Hφ) such that f p = L(χpP ) is the projection onto the closed linearspan of the images of πφ(iX (ξ)) for all ξ ∈ X p. Hence 1− Q = ∨p �=e f p, and thereforeQ is the strong operator limit of the projections

∏p∈J (1 − f p) over the net of finite

subsets J ⊂ P\{e}. Since f p ≥ fq for p ≤ q, it suffices to consider subsets J of anygiven generating set S ⊂ P\{e}. Since

∏

p∈J

(1 − f p) = 1 +∑

∅�=K⊂J

(−1)|K | fqK ,

we therefore get, for all a ∈ A, that

τ0(a) = limJ

(τ(a) +

∑

∅�=K⊂J

(−1)|K |(πφ(a) fqK vφ, vφ)).

In view of equality (2.3) this is exactly (6.2).This proves (6.2) for the elements of A. In order to see that the same is true for M(A)

we just have to observe that all the expressions we considered, like (πφ(a)Qvφ, vφ),(πφ(a) fqK vφ, vφ) and TrqKτ (a), have obvious extensions to M(A) by strict continuity,so the same proof works for M(A).

(ii), (iii) Consider the generalized Gibbs state ψ = TrF(X)τ0 (·D−β

N ) on L(F(X))

discussed in Sect. 2. Then the restrictions of ψ and φ f to K(F(X)) are both KMSβ

with respect to the dynamics Ad DitN , and by the construction of ψ they coincide on

QeK(F(X))Qe = AQe. Since Qe is a full projection in the multiplier algebra ofK(F(X)), it follows that ψ = φ f onK(F(X)) by the discussion at the end of Sect. 1.3.By strict continuity we then have ψ = φ f on L(F(X)). This proves both (ii) and (iii).��Remark 6.9. Since the trace τ0 is the weak∗ limit of a decreasing net of positive traces onthe unital C∗-algebra M(A), it is actually the norm limit of these traces. The above proofmakes this particularly transparent, as it hinges on the norm convergence

∏p∈J (1 −

f p)vφ −→J

Qvφ .


Corollary 6.10. In the setting of Theorem 6.8, let φ be a σ -KMSβ -state onNT (X) andτ = φ|A. Then(i) φ is of infinite type if and only if the limit in (6.2) is zero for all a ∈ A, or equivalently,

for a = 1;(ii) φ is of finite type if and only if τ = ∑

p∈P N (p)−βFpτ0 for some positive trace τ0on A; in this case τ0 and φ are uniquely determined by τ : τ0 is given by (6.2) andφ is given by (6.3) (with φ = φ f ).

Proof. (i) The state φ is of infinite type if and only if φ f = 0, which by part (iii) ofthe theorem is equivalent to τ0 = 0 on A. This proves the “if and only if” part ofthe statement. The last equivalence part follows, since even if A is nonunital, bythe strict continuity of φ f , the functional τ0 = φ f (· Qe) is zero on A if and only ifφ f (Qe) = 0.

(ii) The “only if” statement follows from part (ii) of the theorem. To prove the converse,assume τ = ∑

p∈P N (p)−βFpτ0 for some τ0. Consider the stateψ = TrF(X)τ0 (·D−β

N )

on L(F(X)). Then, on the one hand, we have ψ(aQe) = τ0(a) for a ∈ A byconstruction. On the other hand, ψ ◦ � is a σ -KMSβ -state onNT (X) of finite type,so we can compute ψ(aQe) using part (i) of the theorem. The conclusion is that τ0must be given by (6.2). But then by part (iii) of the theorem we get φ f = ψ ◦ �.Since ψ ◦ � is a state, it follows that φ = φ f , so φ is of finite type and it is givenby (6.3). ��We can now give the following definition.

Definition 6.11. With X , N and β fixed, we say that a positive trace τ on A satisfy-ing (2.1) is of finite type, if τ = ∑

p∈P N (p)−βFpτ0 for some positive trace τ0 on A,which is then necessarily given by (6.2). We say that τ is of infinite type, if the limitin (6.2) is zero for all a ∈ A, equivalently, for a = 1 (with 1 ∈ M(A) and τ(1) = ‖τ‖if A is nonunital).

Note that even with X and N fixed, the same trace can be of finite type for one β andof infinite type for another.

Corollary 6.12. In the setting of Theorem6.8, any tracial state τ on A satisfying (2.1)hasa unique decomposition τ = τ f + τ∞ where τ f is of finite type and τ∞ is of infinite type.Explicitly, we first define a positive trace τ0 on A by (6.2), then τ f = ∑

p∈P N (p)−βFpτ0and τ∞ = τ − τ f .

Proof. By considering the unique gauge-invariant σ -KMSβ -state φ onNT (X) extend-ing τ , this follows from the existence and uniqueness of a decomposition of φ into finiteand infinite type parts and Corollary 6.10. ��

The following is contained in part (ii) of Corollary 6.10, but it is worth stating itexplicitly.

Corollary 6.13. In the setting of Theorem 6.8, we have an affine bijection between theσ -KMSβ -states φ on NT (X) of finite type and the positive traces τ0 on A such that

∑

p∈P

N (p)−β Tr pτ0(1) = 1.

Namely, going from φ to τ0 is by (6.2), and back by (6.3) (with φ = φ f ). Furthermore, ifφ is such a state and ψ is another σ -KMSβ -state (a priori not necessarily of finite type)such that ψ |A = φ|A, then ψ = φ.


7. Traces of Finite Type and Critical Temperature

In practice, checkingwhether a tracial state is of finite type by definition can be a difficultproblem. The following result provides a powerful sufficient condition.

Theorem 7.1. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyaligned product system of C∗-correspondences over P, with Xe = A. Consider thedynamics (1.7) on NT (X) defined by a homomorphism N : P → (0,+∞). Fix β ∈ R

and assume that τ is a tracial state on A satisfying (2.1) such that

∑

p∈P

N (p)−β Tr pτ (1) < ∞.

Then τ is of finite type.

For P = Zn+ this theorem is an immediate consequenceof thefiniteness criterion (3.3).

But this case is quite special. For general P the proof is based on the following lemmaof independent interest.

Lemma 7.2. Assume (G, P) is a quasi-lattice ordered group and μ is a finitely additiveprobability measure on (P,BP ) such that

∑

p∈P

μ(pP) < ∞.

For every p ∈ P, define wp = inf{μ(�) | p ∈ � ∈ BP }. Then, for every � ∈ BP , wehave

μ(�) =∑

p∈�

wp.

In particular, μ extends to a σ -additive probability measure defined on all subsets of P.

Proof. The key part is to establish that

∑

p∈P

wp ≥ 1. (7.1)

In order to prove this inequality, take ε > 0 and choose a finite set F ⊂ P such that∑p/∈F μ(pP) < ε. We claim that if for every p ∈ F we take a set �p ∈ BP containing

p, then

∑

p∈Fμ(�p) > 1 − ε. (7.2)

This implies that∑

p∈F wp ≥ 1 − ε, and since ε could be taken arbitrarily small,proves (7.1).

To prove (7.2), consider the set ∩p∈F (P\�p). By Lemma 2.4(i) it can be written asthe disjoint union of finitely many sets of the form qP or ∩r∈J q(P\r P), with q ∈ Pand J ⊂ P\{e}. It is clear that for every such set appearing in the disjoint union theelement q does not lie in any of the sets �p, hence q /∈ F . Therefore if E is the set of


such q’s, then E is finite, E ∩ F = ∅ and P is covered by the sets �p (p ∈ F) and qP(q ∈ E). It follows that

1 ≤∑

p∈Fμ(�p) +

∑

q∈Eμ(qP) <

∑

p∈Fμ(�p) + ε.

This proves (7.2) and hence (7.1).Next, we claim that

∑

p∈�

wp ≤ μ(�) for any � ∈ BP . (7.3)

Indeed, take any finite subset F ⊂ � and disjoint sets�p (p ∈ F) such that p ∈ �p ⊂ �

for all p ∈ F . This is possible as BP separates points. Then

∑

p∈Fwp ≤

∑

p∈Fμ(�p) ≤ μ(�),

which is what we need.Now, from (7.1) and (7.3), for every � ∈ BP , we get

1 ≤∑

p∈P

wp =∑

p∈�

wp +∑

p∈P\�wp ≤ μ(�) + μ(P\�) = 1,

hence∑

p∈� wp = μ(�). ��Proof of Theorem 7.1. Let τ0 be the trace on A given by (6.2). We have to show that

τ(a) =∑

p∈P

N (p)−β Tr pτ0(a) (7.4)

for all a ∈ A.Fix a ∈ A+ and consider the measure μa on (P,BP ) defined by (2.5). Since by

definition we have μa(pP) = N (p)−β Tr pτ (a), we can apply Lemma 7.2 and concludethat, with wp = inf{μa(�) | p ∈ � ∈ BP }, we have ∑

p∈P wp = τ(a). We claim thatthis is exactly identity (7.4), that is,

wp = N (p)−β Tr pτ0(a) for all p ∈ P. (7.5)

In order to prove (7.5), for every finite subset J ⊂ P\{e} consider the positive traceτJ on A defined by

τJ = τ +∑

∅�=K⊂J

(−1)|K |N (qK )−βFqK τ.

With the sets J partially ordered by inclusion, the traces τJ converge in the weak∗topology (and even in norm) to τ0. By (2.6) we also have

μa

( ⋂

q∈J

p(P\qP))

= N (p)−β(FpτJ )(a).


On the other hand, by Lemma 2.4(ii) we know that every set inBP containing p containsa set of the form ∩q∈J p(P\qP), so in computing wp it suffices to take the limit oversuch sets. In other words, we have

wp = limJ

N (p)−β(FpτJ )(a),

Therefore in order to establish (7.5) we have to show that FpτJ → Fpτ0 in the weak∗topology for all p ∈ P . But this is true by Lemma 1.11(ii), since τJ ≤ τ for all finiteJ ⊂ P\{e}. ��

Combining this theorem with Corollary 6.13 we get the following.

Corollary 7.3. In the setting of Theorem 7.1, assume that for some β ∈ R we have∑

p∈P

N (p)−β Tr pτ (1) < ∞ for all tracial states τ on A. (7.6)

Then all σ -KMSβ -states on NT (X) are of finite type, so we get an affine bijectionbetween the σ -KMSβ -states on NT (X) and the positive traces τ0 on A such that∑

p∈P N (p)−β Tr pτ0(1) = 1.

Note that since by rescaling the traces τ0 as above we get all tracial states on A,we can also say that we have a bijective correspondence between the σ -KMSβ -stateson NT (X) and the tracial states on A. This correspondence, however, is not affine ingeneral.

Definition 7.4. Assuming that N (p) ≥ 1 for all p ∈ P , we denote by βc the infimumof β ∈ R such that condition (7.6) holds, and call it the critical inverse temperature.

The critical inverse temperature equals the maximum of the abscissas of convergenceof the series

∑p∈P N (p)−β Tr pτ (1), since if we have a sequence of tracial states τn

and an increasing sequence (βn)n of inverse temperatures converging to βc such that∑p∈P N (p)−βn Tr pτn (1) = ∞ for all n, then for the tracial state τ = ∑∞

n=1 2−nτn we

have∑

p∈P N (p)−β Tr pτ (1) = ∞ for all β < βc.In this terminology, Corollary 7.3 contains the following statement.

Corollary 7.5. In the setting of Theorem 7.1, assume that N (p) ≥ 1 for all p ∈P and that the critical inverse temperature βc satisfies βc < +∞. Then, for everyβ > βc, all σ -KMSβ -states on NT (X) are of finite type, so we get an affine bijec-tion between the σ -KMSβ -states on NT (X) and the positive traces τ0 on A such that∑

p∈P N (p)−β Tr pτ0(1) = 1.

Let us now discuss assumption (7.6) in more detail. Recall from Sect. 2 that wedenote by T (A) ⊂ A∗ the linear span of tracial states. If the trace Fpτ is finite for anytracial state τ , then Fp extends by linearity to an operator on T (A). Furthermore, sincethe operator Fp is positive, standard arguments show that it must then be bounded.

The only realistic general assumption that guarantees boundedness of Fp seems tobe that each right C∗-Hilbert A-module X p is isomorphic to a direct summand of Am(p)

for some m(p) ∈ N. Then ‖Fp‖ ≤ m(p).

Remark 7.6. If A is unital, a right C∗-Hilbert A-module Y is isomorphic to a directsummand of Am if and only if Y is finitely generated, and the smallest such m is theminimal number of generators, equivalently, the minimal cardinality of a Parseval frame


in Y . Briefly, this can be seen as follows. Assume Y is generated by ξ1, . . . , ξm , so thatwe have a surjective A-modulemap S : Am → Y , Sei = ξi , where (ei )mi=1 is the standardframe in Am . Then S is adjointable, so by [23, Theorem 3.2] the submodule ker S ⊂ Am

is complemented. It follows that Y is isomorphic, hence isometrically isomorphic, to acomplemented submodule of Am , and the projection of (ei )mi=1 onto this submodule isa Parseval frame of cardinality m.

Therefore, although formally Corollary 7.3 does not require anything special aboutthe product system X , in practice it mostly applies to the “finite rank” case. For suchsystems the following result shows that, under additional assumptions,we see a change inthe behavior ofKMS-states atβc, justifying the name “critical inverse temperature”. Thisresult is partly motivated by [28, Theorem 2.5] and generalizes [4, Proposition 4.5(1)],and its proof uses ideas from both papers.

Proposition 7.7. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyaligned product system of C∗-correspondences over P, with Xe = A. Consider thedynamics (1.7) on NT (X) defined by a homomorphism N : P → (0,+∞). Assume inaddition that

(a) the monoid P has a finite generating set S ⊂ P\{e};(b) N (p) > 1 for all p ∈ P\{e};(c) A is unital;(d) the right C∗-Hilbert A-modules X p are finitely generated.

Then

(i) the critical inverse temperature βc is the smallest number, possibly −∞, such thatthe operator

1 +∑

∅�=K⊂S

(−1)|K |N (qK )−βFqK

on T (A) is invertible for all β > βc;(ii) if βc �= −∞, then there exists a tracial state τ on A which satisfies (2.1) for β = βc

and is of infinite type; in particular,∑

p∈P N (p)−βc Tr pτ (1) = ∞.

Proof. Consider the operators Sβ and Tβ on T (A) defined by

Sβ =∑

p∈P

N (p)−βFp, Tβ = 1 +∑

∅�=K⊂S

(−1)|K |N (qK )−βFqK .

The operator Tβ is well-defined for all real β, as well as for complex ones, in whichcase we write Tz to reserve β for the real numbers. The operator Sβ is well-defined forβ > βc, if the convergence of the series above is understood pointwise. Note that theseries does converge absolutely for β large enough, since P is assumed to be finitelygenerated and so an upper bound for

∑p∈P N (p)−β‖Fp‖ can be obtained by looking

at the free monoid F+S with generators gs , s ∈ S, and the series∑

g=gs1 ...gsn∈F+S

N (s1)−β . . . N (sn)

−β‖Fs1‖ . . . ‖Fsn‖.

By Corollary 6.10(ii), for β > βc, we have Tβ Sβτ0 = τ0 for any positive trace τ0 onA (this is also easy to see directly), hence for all τ0 ∈ T (A). Since by assumption (b)


we have N (qK ) > 1 for all ∅ �= K ⊂ S with qK < ∞, for large β the operator Tβ isclose to 1 and hence is invertible. It follows that Sβ = T−1

β for all β large enough.Let now β1 be the smallest number, possibly −∞, such that Tβ is invertible for all

β > β1. Let β2 ≥ β1 be the smallest number such that Sβ is well-defined and Sβ = T−1β

for all β > β2. We claim that β2 = β1.Assume this is not true. Then there is ε > 0 such that β2−ε > β1 and Tz is invertible

for z in the disc D = {z : |z − β2| < ε}. Since Tz is analytic in z, the operator-valuedmap D � z �→ T−1

z is analytic. Take a positive trace τ on A and a ∈ A+. Thenthe Dirichlet series

∑p∈P N (p)−β(Fpτ)(a) converges for all β > β2 and extends by

analyticity to the function D � z �→ (T−1z τ)(a). By the proof of Landau’s theorem,

see [13, Theorem 10], it follows that the Dirichlet series converges to this function forall β ∈ D ∩ R. Therefore the series

∑p N (p)−βFpτ converges in the weak∗ topology,

hence in norm, to T−1β τ for all β > β2 − ε. Since this is true for all τ , this contradicts

the definition of β2. Thus β2 = β1. In particular, βc ≤ β1.If β1 = −∞, there is nothing left to prove. So assume β1 is finite. Then Tβ1 is not

invertible, as otherwise Tβ would also be invertible for β in a neighbourhood of β1,which is not possible. Take a sequence of real numbers tn such that tn ↓ β1. We claimthat ‖T−1

tn ‖ → ∞.Assume this is not true. Then by passing to a subsequence we may assume that the

sequence (T−1tn )n is bounded. But then the identity

T−1tn − T−1

tm = T−1tn (Ttm − Ttn )T

−1tm

shows that (T−1tn )n is a Cauchy sequence converging to an operator S, and by passing to

the limit in the identities Ttn T−1tn = T−1

tn Ttn = 1 we get Tβ1 S = STβ1 = 1, which is acontradiction.

By the uniform boundedness principle it follows then that there exists τ0 ∈ T (A)

such that the sequence (‖T−1tn τ0‖)n is unbounded. Since T (A) is spanned by positive

traces, we may assume that τ0 is a tracial state. By passing to a subsequence we mayalso assume that ‖T−1

tn τ0‖ → ∞. As (Stnτ0)(1) = (T−1tn τ0)(1) = ‖T−1

tn τ0‖, by lettingn → ∞ we conclude that

∑p∈P N (p)−β1(Fpτ0)(1) = ∞. Hence βc = β1, which

completes the proof of part (i).Next, consider the tracial states

τn = T−1tn τ0

‖T−1tn τ0‖

.

Let τ be a weak∗ cluster point of (τn)n . Since A is unital, τ is a tracial state. SinceT−1tn = Stn = ∑

p∈P N (p)−tn Fp, the trace τn satisfies condition (2.1) for β = tn . Sincethe right C∗-Hilbert A-modules X p are finitely generated and therefore admit finiteParseval frames, the operators Fp are obviously weakly∗ continuous. From this we mayconclude that τ satisfies condition (2.1) for β = βc.

Since Ttnτn = ‖T−1tn τ0‖−1τ0, by passing to the limit we see also that Tβcτ = 0,

that is, the trace τ is of infinite type and therefore∑

p∈P N (p)−βc Tr pτ (1) = ∞ byTheorem 7.1. This proves part (ii) of the proposition. ��

From the proof of the proposition we get the following result complementing Corol-lary 7.5.


Corollary 7.8. In the setting of Proposition 7.7, for every β > βc, the map φ �→ φ|Adefines a one-to-one correspondence between the σ -KMSβ -states on NT (X) and thetracial states τ on A such that

τ +∑

∅�=K⊂S

(−1)|K |N (qK )−βFqK τ ≥ 0. (7.7)

In particular, we see that for β > βc condition (2.1) reduces to one inequality corre-sponding to J = S.

Proof. In the notation from the proof of Proposition 7.7, Corollary 7.5 states that themap φ �→ φ|A defines a one-to-one correspondence between the σ -KMSβ -states onNT (X) and the tracial states on A of the form Sβτ0, with τ0 ≥ 0. On the other hand,condition (7.7) means that Tβτ ≥ 0. Therefore we have to prove that a tracial state τ

equals Sβτ0 for a positive trace τ0 if and only if the trace Tβτ is positive. But this isclear, since Sβ = T−1

β . ��Example 7.9. In the setting of Proposition 7.7 consider P = Z

n+. In this case the operator

from part (i) of the proposition is

n∏

i=1

(1 − N (ei )−βFi ),

where, as before, Fi = Fei . Therefore βc is the largest number such that N (ei )βc belongsto the spectrum of Fi for some i . By [28, Theorem 2.5] we know that the largest positiveeigenvalue of Fi coincides with the spectral radius of Fi . (To be more precise, the resultin [28] is formulated for a more restricted class of operators, but its proof works for anyweakly∗ continuous positive operator on T (A).) We conclude that

βc = max1≤i≤n

log r(Fi )

log N (ei ), (7.8)

where r(Fi ) denotes the spectral radius of Fi .This equality shows that whenever r(Fi ) > 1 for all i , the most natural dynamics on

NT (X) is given by N (ei ) = r(Fi ). This type of dynamics was first singled out in [31]and is now called the preferred dynamics [15].

Returning to the more general setting of Corollary 7.3, observe that the uniformboundedness principle implies that, under the assumptions of that corollary, the partialsums

∑p∈F N (p)−βFp are uniformly bounded when F runs over finite subsets of P .

We do not know whether this implies that the series

∑

p∈P

N (p)−β‖Fp‖ (7.9)

is convergent. Similarly,wedonot knowwhether the critical inverse temperature (definedif N (p) ≥ 1 for all p) coincides with the abscissa of convergence of the series (7.9)in general. Equality (7.8) implies that these two numbers do coincide for the monoidsP = Z

n+ under the assumptions of Proposition 7.7.


Remark 7.10. Series (7.9) can be convergent for nontrivial reasons only under someadditional assumptions on P . For instance, if we assume that Fp �= 0 for all p (andthis is the case, e.g., if every right C∗-Hilbert module X p is full and T (A) �= 0), thenfiniteness of (7.9) for some β ∈ R implies that P has finite initial intervals {q | q ≤ p}.Indeed, otherwise there exist p, qn and rn , such that p = qnrn and the elements qn areall different. Then

N (p)−β‖Fp‖ ≤ N (qn)−β‖Fqn‖N (rn)

−β‖Frn‖,and since both N (qn)−β‖Fqn‖ and N (rn)−β‖Frn‖ must converge to zero, we get acontradiction.

8. Gauge-Invariance

One of the obvious shortcomings of Theorem5.1 is that it deals onlywith gauge-invariantKMS-states. By the results of the previous section we know, however, that for large β

all KMSβ -states can happen to be of finite type and, in particular, gauge-invariant. Thefollowing generalization of [4, Proposition 2.2] can sometimes be used to establishgauge-invariance of KMS-states of infinite type.

Proposition 8.1. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyaligned product system of C∗-correspondences over P, with Xe = A. Consider thedynamics (1.7) onNT (X) defined by a homomorphism N : P → (0,+∞). Given a σ -KMSβ -state φ for some β ∈ R, consider the trace τ = φ|A and assume that it satisfiesthe following property: for any strictly increasing sequence {pn}∞n=1 in P, we have

limn→∞ N (pn)

−β Tr pnτ (1) = 0. (8.1)

Then φ is gauge-invariant.

Proof. In order to make the argument more transparent, let us assume first that A isunital and the right C∗-Hilbert A-modules X p admit finite Parseval frames. For every

p ∈ P , we fix such a frame (λ(p)i )i and define a completely positive map

Sp : NT (X) → NT (X) by Sp(x) =∑

i

iX (λ(p)i )∗x iX (λ

(p)i ).

Then the KMS-state φ has the following property:

if p ∨ q < ∞ and x ∈ iX (X p)∗iX (Xq),

then Sp−1(p∨q)(x) ∈ iX (Xq−1(p∨q))∗iX (X p−1(p∨q))

and φ(x) = N (p−1(p ∨ q))−βφ(Sp−1(p∨q)(x)). (8.2)

Indeed, it suffices to consider x = iX (ξ)∗iX (ζ ), with ξ ∈ X p and ζ ∈ Xq . Then

the first statement is clear as, writing λi instead of λ(p−1(p∨q))i for simplicity, we have,

using (1.3), that

iX (λi )∗iX (ξ)∗iX (ζ ) = iX (�(ζ )∗(ξλi ))

∗ and �(ζ )∗(ξλi ) ∈ Xq−1(p∨q).


For the second statement note that since by (1.5) the element iX (ξ)∗iX (ζ ) can be writtenas the sum of elements iX (η)iX (μ)∗, with η ∈ X p−1(p∨q) and μ ∈ Xq−1(p∨q), and

∑

i

iX (λi )iX (λi )∗iX (η) = iX (η),

we have x = ∑i iX (λi )iX (λi )

∗x . Therefore by applying φ and the KMS-condition weget

φ(x) =∑

i

φ(iX (λi )iX (λi )∗x) = N (p−1(p ∨ q))−β

∑

i

φ(iX (λi )∗x iX (λi )),

which proves (8.2).Now, in order to prove that φ is gauge-invariant, we have to show that given p �= q

and ξ ∈ X p, ζ ∈ Xq , we have φ(iX (ζ )iX (ξ)∗) = 0. Applying the KMS-condition weget

φ(iX (ζ )iX (ξ)∗) = N (q)−βφ(iX (ξ)∗iX (ζ )).

If p∨ q = ∞, we are done, since then iX (ξ)∗iX (ζ ) = 0 by Nica covariance. Otherwisewe let x = iX (ξ)∗iX (ζ ) and proceed as follows.

Put p1 = q−1(p ∨ q) and q1 = p−1(p ∨ q). If p �= p ∨ q, we define r1 = q1 andby (8.2) get

Sr1(x) ∈ iX (X p1)∗iX (Xq1) and φ(x) = N (r1)

−βφ(Sr1(x)). (8.3)

If p = p ∨ q, then we must have q �= p ∨ q. In this case we define r1 = p1, andapplying (8.2) to x∗ = iX (ζ )∗iX (ξ) instead of x we again get (8.3). Then we apply thesame procedure to Sr1(x) instead of x , and so on.

There are two possibilities. One is that this process continues indefinitely. Thenafter n steps we get elements p1, . . . , pn, q1, . . . , qn, r1, . . . , rn ∈ P , with pk �= qk ,rk ∈ {pk, qk}\{e} and pk ∨ qk < ∞ for all k, such that (Srn ◦ · · · ◦ Sr1)(x) lies iniX (X pn )

∗iX (Xqn ) and

φ(x) = N (r1)−β . . . N (rn)

−βφ((Srn ◦ · · · ◦ Sr1)(x)). (8.4)

Since the positive linear functional φ ◦ Srn ◦ · · · ◦ Sr1 has norm

φ((Srn ◦ · · · ◦ Sr1)(1)) = Trr1...rnτ (1),

we then get

|φ(x)| ≤ N (r1 . . . rn)−β Trr1...rnτ (1)‖x‖.

Since the sequence {r1 . . . rn}∞n=1 is strictly increasing, lettingn → ∞, by the assumptionof the propositionwe obtainφ(iX (ξ)∗iX (ζ )) = φ(x) = 0, hence alsoφ(iX (ζ )iX (ξ)∗) =0.

The other possibility is that the process stops after n steps for some n ≥ 1, thatis, we get pn ∨ qn = ∞. Then iX (X pn )

∗iX (Xqn ) = 0 by Nica covariance, hence(Srn ◦ · · · ◦ Sr1)(x) = 0, so we again get φ(x) = 0. This finishes the proof of theproposition under the additional assumption that A is unital and the right A-modules X pare finitely generated.


In the general case the proof is basically the same, but at every step instead of exactidentities we get approximate ones. Namely, for every p ∈ P , choose an approximateunit (u(p)

i )i∈Ip inK(X p) of the form u(p)i = ∑

ξ∈J (p)i

θξ,ξ for some finite sets J (p)i ⊂ X p.

Then define

S(i)p : NT (X) → NT (X) by S(i)

p (x) =∑

ξ∈J (p)i

iX (ξ)∗x iX (ξ).

Following the proof of (8.2) we then get, for x ∈ iX (X p)∗iX (Xq), that

S(i)p−1(p∨q)

(x) ∈ iX (Xq−1(p∨q))∗iX (X p−1(p∨q)),

φ(i (p−1(p∨q))

X (u(p−1(p∨q))i )x) = N (p−1(p ∨ q))−βφ(S(i)

p−1(p∨q)(x)),

and i (p−1(p∨q))

X (u(p−1(p∨q))i )x −→

ix . Correspondingly, instead of (8.3) we get

S(i1)r1 (x) ∈ iX (X p1)

∗iX (Xq1) and |φ(x) − N (r1)−βφ(S(i1)

r1 (x))| <ε

2,

where ε > 0 is any fixed number and i1 is a sufficiently large index in Ir1 . At the nextstep we replace ε by ε/2, and so on. Then after n steps instead of (8.4) we get in additionto pk, qk, rk indices ik such that

|φ(x) − N (r1)−β . . . N (rn)

−βφ((S(in)rn ◦ · · · ◦ S(i1)

r1 )(x))| <

n∑

k=1

ε

2k.

Since ‖φ◦S(in)rn ◦· · ·◦S(i)

r1 || ≤ Trr1...rnτ (1), this is still enough to conclude that |φ(x)| ≤ ε,and as ε > 0 was arbitrary, we get φ(x) = 0. ��Remark 8.2. Since any strictly increasing sequence in P tends to infinity (in the usualsense, that is, it eventually leaves every finite subset of P), the condition

limp→∞ N (p)−β Tr pτ (1) = 0 (8.5)

is stronger than (8.1). Observe next that if P has finite initial intervals (recall that byRemark (7.10) this condition is automatically satisfied if Fp �= 0 for all p and theseries (7.9) is convergent for some β), then (8.5) is satisfied for any trace τ of finite type.Indeed, when p → ∞, the sets pP eventually do not intersect any given finite subsetof P , so if τ = ∑

q∈P N (q)−βFqτ0, then

N (p)−β Tr pτ (1) =∑

q∈pP

N (q)−β Trqτ0(1) → 0.

Therefore, under a very mild additional assumption, Proposition 8.1 gives an alternativeproof of the fact that if τ is of finite type, then it can only be extended to a gauge-invariantKMS-state on NT (X).


Corollary 8.3. Assume (G, P) is a quasi-lattice ordered group and X is a compactlyalignedproduct systemofC∗-correspondences over P,with Xe = A. Suppose in additionthat A is unital and X p is finitely generated as a right A-module for all p ∈ P. Considerthe dynamics (1.7) on NT (X) defined by a homomorphism N : P → (0,+∞), andassume that for some β �= 0 and δ > 0 we have

N (p)β ≥ (1 + δ)m(p) (8.6)

for all p ∈ P\{e}, where m(p) is the minimal number of generators of X p as a rightA-module. Then every σ -KMSβ -state on NT (X) is gauge-invariant.

Recall that by Remark 7.6 the number m(p) equals the minimal cardinality of aParseval frame in X p.

Proof. We claim that condition (8.1) is satisfied for any tracial state τ on A. It is clearfrom the definition of induced traces that if a right C∗-Hilbert A-module Y has a Parsevalframe consisting of m elements, then TrYτ (1) ≤ m. Hence, for any r1, . . . , rn ∈ P\{e},we have

Trr1...rnτ (1) ≤ m(r1) . . .m(rn) ≤ (1 + δ)−nN (r1 . . . rn)β,

from which we see that (8.1) is satisfied. ��Remark 8.4. The function m is submultiplicative on P , that is, m(pq) ≤ m(p)m(q) forp, q ∈ P . It follows that (8.6) is satisfied for all p ∈ P\{e} once it is satisfied for all pin a generating set of P . In particular, if P is finitely generated and N (p) > 1 for all pin a finite generating set, then (8.6) is satisfied for all β large enough. Such β’s can stillbe smaller than βc, see [4] and Example 9.6 below.

Remark 8.5. Although Corollary 8.3 involves assumptions on N , it still has a ratherstraightforward generalization to arbitrary quasi-free dynamics. Namely, the analoguesof the maps N (p)−β S(i)

p from the proof of Proposition 8.1 are

S(i)p (x) =

∑

ξ∈J (p)i

iX (U (p)iβ/2ξ)∗x iX (U (p)

iβ/2ξ),

where J (p)i ⊂ X p is a finite set of vectors analytic with respect to the one-parameter

group (U (p)t )t (and such that

∑ξ∈J (p)

iθξ,ξ ≤ 1). There are two ways of getting an

estimate for φ(S(i)p (x)). One is to use that by definition we have (φ ◦ S(i)

p )(1) ≤(IndU

(p)

X p(φ|A)

)(1). This leads to an analogue of Proposition 8.1. Another is to use that

‖S(i)p ‖ ≤ |J (p)

i | ‖U (p)iβ/2‖2. This leads to an analogue of Corollary 8.3 where (8.6) gets

replaced by the condition ‖U (p)iβ/2‖−2 ≥ (1 + δ)m(p).

9. Right-Angled Artin Monoids

In this section we will show that for a class of groups interpolating between free groupsand free abelian groups condition (2.1) reduces to a much smaller system of inequalities.


Let� be a simple graph, finite or infinite. The right-angled Artin group G� associatedto � is a group with generators si indexed by the vertices of � and with relations

si s j = s j si , if i and j are connected by an edge.

We denote by S� the set of the generators si and call it the standard generating set ofG� . Denote by P� ⊂ G� the monoid generated by S� . It is shown in [8] that (G�, P�)

is a quasi-lattice ordered group.

Theorem 9.1. Consider the quasi-lattice ordered right-angled Artin group (G�, P�)

associated with a simple graph �. Assume we are given a product system X of C∗-correspondences over P� , with Xe = A, a homomorphism N : P� → (0,+∞) andβ ∈ R. Then a tracial state τ on A satisfies condition (2.1) if and only if

τ(a) +∑

K∈C(S�):∅�=K⊂J

(−1)|K |N (sK )−β TrsKτ (a) ≥ 0 for all finite J ⊂ S� and a ∈ A+,

(9.1)

where C(S�) is the collection of finite sets of pairwise commuting standard generators,or in other words, the collection of sets of generators corresponding to complete finitesubgraphs of �, and sK = ∏

s∈K s.

For the proof we need the following partial description of the operation ∨ on P� ,which is a simple consequence of [8, Proposition 13].

Lemma 9.2. For any p = si1 . . . sin ∈ P� and si ∈ S� , we have

(a) if i ∈ {i1, . . . , in} and for the lowest index k such that i = ik the vertex i and eachvertex i j with j < k are connected by an edge, then p ∨ si = p;

(b) if i /∈ {i1, . . . , in}, but the vertex i and each vertex i j are connected by an edge, thenp ∨ si = psi = si p;

(c) if neither (a) nor (b) applies, then p ∨ si = ∞.

An immediate consequence of this lemma is that if K ⊂ S� is a finite set of generators,then

qK =∨

s∈Ks =

{sK , if K ∈ C(S�),

∞, otherwise.(9.2)

Lemma 9.3. Let (G, P) = (G�, P�). Then every set � ∈ BP can be written as a finitedisjoint union of the sets pP and p(∩s∈J (P\sP)), where p ∈ P and J ⊂ S� is a finitenonempty set.

Proof. By Lemma 2.4(i) we know that � can be written as a finite disjoint union of thesets pP and p(∩q∈J (P\qP)), where J ⊂ P\{e} is finite. In the first case there is nothingto prove. In the second case each set P\qP can be further decomposed as follows.Writeq as sk1 . . . skm . Then P\qP is the disjoint union of the sets sk1 . . . ski−1 P\sk1 . . . ski Pfor i = 1, . . . ,m. It follows that every set p(∩q∈J (P\qP)), where J ⊂ P\{e} is finite,can be written as the disjoint union of sets of the form

p(q1P\q1si1 P) ∩ · · · ∩ p(qn P\qnsin P).


With sets of the latter form we proceed as follows. We may assume that q = q1 ∨· · · ∨ qn is finite, as otherwise we get the empty set. Then the above set is contained inpqP , so we can write it as

p(qP\q1si1 P) ∩ · · · ∩ p(qP\qnsin P). (9.3)

Now, for every j = 1, . . . , n, put r j = q−1j q. Then

p(qP\q j si j P) = pq j (r j P\si j P) = pq j (r j P\(r j ∨ si j )P),

with the convention ∞P = ∅. But by Lemma 9.2 we have only three options for the set(r j∨si j )P: r j P, r j si j P, ∅. Correspondingly, the only options for the set p(qP\q j si j P)

are

∅, pq(P\si j P), pqP.

Hence the set (9.3) is either empty, pq(∩s∈J (P\sP)), where J ⊂ {si1, . . . , sin }, orpqP . ��Proof of Theorem 9.1. Denote P� by P . From (9.2)we see that condition (9.1) is nothingother than condition (2.1) applied to the sets J ⊂ S� . Thus (2.1) implies (9.1).

In order to prove the converse, observe first of all that if (9.1) is satisfied, then Fsτis finite and dominated by a scalar multiple of τ for all s ∈ S� , hence Tr

pτ is finite for

all p ∈ P by (1.14).Consider now, as in Sect. 2, the T (A)-valued finitely additive measure μ on (P,BP )

defined by μ(pP) = N (p)−βFpτ . By assumption we have μ(�J ) ≥ 0 for all finiteJ ⊂ S� , where �J = ∩s∈J (P\sP). But then, by (2.6), we have

μ(p�J ) = N (p)−βFpμ(�J ) ≥ 0

for all p ∈ P and finite J ⊂ S� . By Lemma 9.3 we conclude that μ is positive, that is,condition (2.1) is satisfied. ��Remark 9.4. When � is a finite complete graph, so that P� = Z

n+, Theorem 9.1 gives

another proof of equivalence of conditions (2.1) and (3.2), whichwas promised in Sect. 3.When � is a finite graph with no edges, so that P� is a free monoid F

+n , then Theo-

rem 9.1 implies that condition (2.1) is equivalent to

n∑

i=1

N (si )−βFiτ ≤ τ,

where Fi = Fsi . However, a product system over a free monoid is determined byone graded correspondence ⊕i Xsi , so Theorem 5.1 and the equivalence of the aboveinequality to (2.1) in this case follows already from [21].

The following corollary shows that for the right-angled Artin monoids the set ofpossible temperatures of KMS-states on NT (X) is often a half-line.

Corollary 9.5. In the setting of Theorem 9.1, assume that N (p) ≥ 1 for all p ∈ P� andτ is a tracial state on A satisfying (2.1) for some β = β0. Then τ satisfies (2.1) for allβ > β0.


Proof. Denote P� by P . We will prove the following statement: if N and N ′ are twohomomorphisms P → (0,+∞) such that N ′(p) ≤ N (p) for all p ∈ P , and τ sat-isfies (9.1) for N and β = −1, then it also satisfies (9.1) for N ′ and β = −1. Thecorollary then follows by applying this to the homomorphisms N−β0 and N−β andusing that conditions (9.1) and (2.1) are equivalent.

The argument is analogous to the one used in the proof of Theorem 3.1 for a similarstatement, but becomes slightly more complicated in the general case of right-angledArtin monoids.

Define a map η from the linear span of the characteristic functions χpP , p ∈ P , intoT (A) by letting η(χpP ) = Fpτ . In view of (9.2), condition (9.1) for N and β = −1 canbe written as

η( ∏

s∈J

(1 − N (s)χsP ))

≥ 0

for all finite J ⊂ S� , and we have to show that the same is true with N replaced by N ′.Using that

∏

s∈J

(1 − N ′(s)χsP ) =∏

s∈J

((1 − N (s)χsP ) + (N (s) − N ′(s))χsP ),

we see that we can write the function∏

s∈J (1− N ′(s)χsP ) as a linear combination withpositive coefficients of functions of the form

( ∏

s∈EχsP

)( ∏

t∈F(1 − N (t)χt P)

),

where E and F are finite disjoint sets of generators. If E or F are empty, then η ispositive on such a function by our assumptions. Otherwise, by (9.2), we may assumethat the elements of E commute with each other. Using (9.2) again, we see also that ifan element of F does not commute with an element of E , then the factor 1 − N (t)χt Pcan be replaced by 1. Therefore we may assume that the elements of E commute witheach other and with the elements of F . Then the above function equals

χsE P +∑

K∈C(S�):∅�=K⊂F

(−1)|K |N (sK )χsE sK P ,

from which we see that the value of η on this function is

FsE τ +∑

K∈C(S�):∅�=K⊂F

(−1)|K |N (sK )−βFsEsK τ = FsEη( ∏

t∈F(1 − N (t)χt P)

)≥ 0.

This completes the proof of the corollary. ��As an immediate application let us consider the following example.

Example 9.6. Let (G, P) be a quasi-lattice ordered group such that P has a finite gen-erating set S ⊂ P\{e}. Assume we are given a homomorphism N : P → (0,+∞)

such that N (p) > 1 for all p ∈ P\{e} and the abscissa of convergence of the series∑p∈P N (p)−β satisfies βc > 0. Consider the full semigroup C∗-algebra of P , that is,

the algebra NT (X) for the trivial product system X p = C, and the dynamics σ on itdefined by N . Then, using our terminology, Theorem 3.5 and Proposition 4.5 in [4] canbe formulated as follows:


(i) for every β > βc there is a unique σ -KMSβ -state, and it is of finite type;(ii) there is a unique σ -KMSβc -state, and it is of infinite type;(iii) if for some β ∈ [0, βc) there is a σ -KMSβ -state, then it is of infinite type.

We remark that parts (i) and (iii) follow also from our results in Sect. 6. Indeed, by thedefinition of βc, the unique tracial state on A = C is of finite type for β > βc, so we get(i) by Corollary 6.13, and there are no traces of finite type for β < βc, so we get (iii).Part (ii) also follows from our Proposition 7.7 and Corollary 8.3, but the proofs of thoseresults use ideas from [4] and do not provide any new perspective here.

By Corollary 6.10(i), part (iii) above means that β must satisfy the equation

1 +∑

∅�=K⊂S

(−1)|K |N (qK )−β = 0,

an observation which was already made in [4]. This implies that the set of possible β’sin (iii) is finite.

Consider now the case when (G, P) = (G�, P�) for a finite simple graph �. Theassumption βc > 0 means precisely that G is not abelian, that is, � is not a completegraph. By Corollary 9.5 we know that the set of possible inverse temperatures is a half-line. We can therefore conclude that there are no σ -KMSβ -states on the full semigroupC∗-algebra of P� for β < βc, so for such systems the classification of KMS-statesreduces completely to the computation of βc.

10. Direct Products of Quasi-Lattice Ordered Monoids

Throughout this sectionwefixaquasi-lattice orderedgroup (G, P)of the form (G1, P1)×· · · × (Gn, Pn), a compactly aligned product system X of C∗-correspondences over P ,with Xe = A, a homomorphism N : P → (0,+∞), and β ∈ R. In this setting, de-veloping the ideas in [7,17], we can refine the decomposition of states on NT (X) intopositive functionals of finite and infinite types by considering states that are finite withrespect to some factors and infinite with respect to other.

We need to introduce some notation in order to formulate the precise result.Wheneverconvenient we view Gi as a subgroup of G. For any set F ⊂ {1, . . . , n} we denote byPF the submonoid of P generated by the monoids Pi for i ∈ F , with the conventionP∅ = {e}. Denote by X (F) the product system (X p)p∈PF . When F = {i} we write X (i)

instead of X ({i}), and we use the same convention for various constructions below. Theembeddings X (F) ↪→ X induce ∗-homomorphisms NT (X (F)) → NT (X).

Assume now that we are given a representation π : NT (X) → B(H). Considerthe von Neumann algebra M = π(NT (X))′′. By restriction we get representationsπF : NT (X (F)) → B(H). PutMF = πF (NT (X (F)))′′ ⊂ M . Clearly,MF = ∨i∈FMi .Following Proposition 6.1 and its proof, for every F we consider the projection QF ∈MF onto the space of vacuum vectors for NT (X (F)) and its central support zF ∈ MF .

Proposition 10.1. In the above setting we have:

(i) for every i = 1, . . . , n, the projection Qi commutes with M j for j �= i and theprojection zi is central in M;

(ii) QF = ∏i∈F Qi and zF = ∏

i∈F zi for every set F ⊂ {1, . . . , n}.


Proof. (i) The first statement says that the space Qi H is invariant under Mj . In order toprove this we have to show that if v ∈ Qi H , ξ ∈ X p for some p ∈ Pi and ζ ∈ Xq forsome q ∈ Pj , then

π(iX (ξ))∗π(iX (ζ ))v = 0 and π(iX (ξ))∗π(iX (ζ ))∗v = 0.

The first equality follows from (1.5), since p ∨ q = pq = qp. The second equalityholds since ζ ξ ∈ Xqp = X pq can be approximated by finite sums of elements of theform ξ ′ζ ′, with ξ ′ ∈ X p and ζ ′ ∈ Xq .

For the second statement we have to show that the space zi H is invariant under Mjfor all j �= i . Since this space is the closed linear span of the vectors π(iX (ξ))v, wherev ∈ Qi H and ξ ∈ X p (p ∈ Pi ), this can be proved using arguments similar to the onesabove.

(ii) The first statement is an immediate consequence of (i), since by definition wehave QFH = ∩i∈F Qi H . For the second statement observe that for each i ∈ F wehave

Mi QF H =( ∏

j∈F\{i}Q j

)Mi Qi H =

( ∏

j∈F\{i}Q j

)zi H.

So by applying Mi (i ∈ F) one by one to the space QFH we see that we can generatethe entire space

( ∏i∈F zi

)H . Since this space is invariant under MF , we conclude that

the central support of QF in MF equals∏

i∈F zi . ��Corollary 10.2. Every positive linear functional φ onNT (X) uniquely decomposes as

φ =∑

F⊂{1,...,n}φF ,

where φF is a positive linear functional onNT (X) defining by restriction a functionalonNT (X (i)) which is of finite type for every i ∈ F and of infinite type for every i ∈ Fc.Furthermore, φF defines a functional of finite type on NT (X (F)).

Proof. Theproof is similar to that ofProposition6.6.Consider theGNS-triple (Hφ, πφ, vφ)

defined by φ. Let zi be the central projections in M = π(NT (X))′′ as in the aboveproposition for π = πφ . Define central projections

wF =( ∏

i∈Fzi

)( ∏

j∈Fc

(1 − z j ))

∈ M.

Then the functionals φF = (π(·)wFvφ, vφ) give the desired decomposition. Note thatsince wF ≤ zF , the restriction of φF to NT (X (F)) gives a functional of finite type.

In order to prove the uniqueness, it suffices to show that if ψ ≤ φ and ψ defines afunctional of finite type on NT (X (i)) for i ∈ F and of infinite type for i ∈ Fc, thenψ ≤ φF .

Let x ∈ M ′, 0 ≤ x ≤ 1, be the unique element such thatψ = (π(·)xvφ, vφ). For every

i , put Ki = Mi x1/2vφ . Then as the GNS-triple associated with ψ |NT (X (i)) we can take(Ki , πi |Ki , x

1/2vφ). It follows that if i ∈ F , then x1/2vφ ∈ Ki ⊂ zi Hφ . ApplyingM andusing that zi is a central projection in M , we then conclude that x1/2Hφ ⊂ zi Hφ , hencex ≤ zi . On the other hand, if i ∈ Fc, then the vector zi x1/2vφ defines a functional offinite type onNT (X (i)) dominated byψ , so it must be zero. Hence x1/2vφ ∈ (1−zi )Hφ

and then x ≤ 1 − zi . It follows that x ≤ wF and therefore ψ ≤ φF . ��


Note that the finite/infinite decomposition of φ is then given by

φ f = φ{1,...,n}, φ∞ =∑

F�{1,...,n}φF .

Similarly to Corollary 6.7, from the construction of the functionals φF we see alsothat if φ satisfies the σ -KMSβ condition for some dynamics σ on NT (X), then φFsatisfy this condition as well.

For the dynamics we are interested in, this leads to the following refinement ofCorollary 6.12.

Corollary 10.3. With N : P → (0,+∞) and β ∈ R fixed, every positive trace τ onA = Xe satisfying (2.1) decomposes uniquely as

τ =∑

F⊂{1,...,n}τF ,

where τF is a positive trace satisfying (2.1) for P such that τF is of finite type withrespect to Pi for every i ∈ F and of infinite type for every i ∈ Fc. Furthermore, τF isof finite type with respect to PF .

Let us also observe the following.

Lemma 10.4. In order to verify condition (2.1) for a positive trace τ on A it sufficesto consider subsets J ⊂ P\{e} of the form J = ∏

i Ji , Ji ⊂ Pi .

Proof. As pP = ∏i pi Pi for p = (p1, . . . , pn), every set in BP decomposes into the

disjoint union of sets of the form∏

i �i , with �i ∈ BPi . By Lemma 2.4(i), each set �idecomposes into a disjoint union of the sets pi Pi and pi (∩q∈Ji (Pi\qPi )). The resultnow follows by the same argument as in the proof of Theorem 9.1. ��

Together with Corollary 10.3 this lemma can in principle simplify analysis of tracessatisfying (2.1) for direct product monoids. Namely, we have the following result.

Proposition 10.5. In the above setting, with N : P → (0,+∞) and β ∈ R fixed, assumethat for every subset F ⊂ {1, . . . , n} we are given a positive trace τF,0 on A = Xe suchthat

(a) τF,0 satisfies condition (2.1) for PFc ;(b) τF,0 is of infinite type with respect to Pi for every i ∈ Fc;(c)

∑p∈PF N (p)−β Tr pτF,0(1) < ∞.

Then the trace

τ =∑

F⊂{1,...,n}

∑

p∈PF

N (p)−βFpτF,0

satisfies condition (2.1) for P, and every positive trace satisfying (2.1) is obtained thisway for uniquely defined positive traces τF,0 satisfying conditions (a)-(c).

Proof. Given a collection of traces τF,0 as in the formulation, it is easy to see that, forevery F , the trace

∑p∈PF N (p)−βFpτF,0 is still of infinite type with respect to Pi for

every i ∈ Fc and, using Lemma 10.4 and identity (2.8), that it satisfies condition (2.1)for P . Hence the trace

τ =∑

F⊂{1,...,n}

∑

p∈PF

N (p)−βFpτF,0


satisfies condition (2.1) for P .Conversely, starting with a trace satisfying condition (2.1) for P , we first apply

Corollary 10.3 and get a decomposition τ = ∑F⊂{1,...,n} τF . Then, by Corollary 6.12

applied to the monoids PF , we get uniquely defined positive traces τF,0 such that

τF =∑

p∈PF

N (p)−βFpτF,0.

From formula (6.2) for τF,0 it is easy to see that every trace FpτF,0 (p ∈ PF ) satisfiescondition (2.1) for PFc . It follows then that since τF is infinite with respect to Pi for everyi ∈ Fc, the traces FpτF,0 are infinite with respect to Pi as well. Therefore conditions(a)-(c) are satisfied.

The uniqueness statement follows from Corollaries 10.3 and 6.12. ��The key point of this proposition is that condition (b) might give strong restrictions

on possible traces and be easier to understand since it involves only individual factorsof P .

Example 10.6. Consider P = Zn+. Denote as usual the standard generators of Z

n+ by

e1, . . . , en and write Fi instead of Fei . Condition (b) in Proposition 10.5 for τF,0 meansthat

N (ei )−βFiτF,0 = τF,0

for all i ∈ Fc. For every such trace condition (2.1) is satisfied for ZFc

+ , namely, the lefthand side of (2.1) side is always zero (this is particularly transparent for the equivalentcondition (3.2)). We therefore conclude that if for every F ⊂ {1, . . . , n} we are given apositive trace τF,0 on A such that

(a′) N (ei )−βFiτF,0 = τF,0 for every i ∈ Fc,(b′)

∑p∈Z

F+N (p)−β Tr pτF,0(1) < ∞,

then the trace

τ =∑

F⊂{1,...,n}

∑

p∈ZF+

N (p)−βFpτF,0 (10.1)

satisfies condition (2.1) (or, equivalently, (3.2)) for Zn+, and any positive trace satisfying

this condition is obtained this way for uniquely defined positive traces τF,0 satisfyingconditions (a′) and (b′).

For finite rank product systems over Zn+ this leads to an alternative proof of [17,

Theorem4.4], which asserts that the tracial states τ as in (10.1) are exactly the restrictionsof the gauge-invariant KMSβ -states onNT (X). Moreover, we see that the result extendsto compactly aligned product systems.

11. Some Examples and Applications

In this section we will consider a few more examples that have been recently studiedin the literature and show how our results allow us to recover them, and sometimesstrengthen, in a quick unified way.


11.1. Product systems arising from surjective local homeomorphisms. Let (G, P) bea quasi-lattice ordered group. Assume Pop acts on a compact Hausdorff space Z bysurjective local homeomorphisms h p : Z → Z , so that we have h pq = hq ◦ h p. Forevery p ∈ P we can consider a C∗-correspondence X p over A = C(Z) as in Exam-ple 1.12; in other words, X p is the correspondence associated with the topological graph(Z , Z , id, h p). These correspondences form a product system, with the product definedby

(ξζ )(z) = ξ(z)ζ(h p(z)) for ξ ∈ X p, ζ ∈ Xq .

Since the rightC(Z)-modules X p are finitely generated, the left actions ofC(Z) on themare by generalized compact operators and the product system X is compactly aligned.

By Example 1.12, the corresponding operators Fp : C(Z)∗ → C(Z)∗ coincide withthe dual Ruelle transfer operatorsL∗

p. We thus see that in order to understand the KMSβ -states of NT (X) with respect to the dynamics given by a homomorphism N : P →(0,+∞), and to get a complete classification of such gauge-invariant states, we need tostudy the representation of P on C(Z)∗ given by p �→ N (p)−βL∗

p. Spectral analysisof just one dual transfer operator is already a difficult problem in dynamical systemstheory, but a few things can be said on a general basis.

Consider the case P = Zn+, so X is described by a family of n commuting surjective

local homeomorphisms on Z . In this case we get the following result.

Proposition 11.1. Let h1, . . . , hn be commuting surjective local homeomorphisms of acompact Hausdorff space Z and let X be the corresponding product system over Z

n+

as above. Let σ be the dynamics (1.7) on NT (X) determined by a homomorphismN : Z

n+ → (0,+∞). For each state φ on NT (X), let μφ be the probability measure on

Z such that φ(a) = ∫Z a dμφ for all a ∈ C(Z). Then

(i) for every β ∈ R, the map φ �→ μφ defines a one-to-one correspondence betweenthe gauge-invariant σ -KMSβ -states onNT (X) and the probability measures μ onZ such that

∏

i∈J

(1 − N (ei )

−βL∗i

)μ ≥ 0 (11.1)

for every nonempty subset J ⊂ {1, . . . , n}, where L∗i = L∗

ei ; in particular, there areno σ -KMSβ -states unless N (ei )β ≥ 1 for all i .

Assume in addition that N (ei ) > 1 for all i and put

ri = limj→∞

(maxz∈Z |h− j

i (z)|)1/j , i = 1, . . . , n,

and

βc = max1≤i≤n

log rilog N (ei )

.

Then

(ii) the number βc is the abscissa of convergence of the series∑

p∈Zn+N (p)−β‖L∗

p‖,and for every β > βc all σ -KMSβ -states on NT (X) are of finite type, so there isan affine bijection between such states and the set

�β = {ν : ν is a positive measure on Z and

∑p∈Z

n+N (p)−βL∗

pν is a probability measure};


(iii) at β = βc we have a phase transition, namely, there is a σ -KMSβc -state onNT (X)

of infinite type.

Proof. (i) The first statement follows from Corollary 3.3. For the second one observethat Li (1) ≥ 1, so if N (ei )−βL∗

i μ ≤ μ for some positive measure μ, then we must haveN (ei )−β ≤ 1.

Turning to (ii) and (iii), note that

‖L ji ‖ = ‖L j

i (1)‖ = maxz∈Z |h− j

i (z)|.

This implies that ri is the spectral radius of Li , or equivalently, of L∗i . Hence, by (7.8),

the number βc coincides with the critical inverse temperature defined in Sect. 7. As wasalready remarked there, the critical inverse temperature coincides with the abscissa ofconvergence of the series

∑p∈Z

n+N (p)−β‖L∗

p‖. The rest of (ii) and (iii) follows fromCorollary 7.5 and Proposition 7.7. ��Remark 11.2. The number log ri often coincides with the topological entropy of hi , see[10]. It is also interesting to note that by the proof of [10, Proposition 2.3], without anyextra assumptions on hi , there is a point zi ∈ Z such that ri = lim j→∞ |h− j

i (zi )|1/j .The description of KMSβ -states in part (ii) of the above proposition has been ob-

tained in [1, Theorem 6.1] under the additional assumptions that the homeomorphismsh1, . . . , hn ∗-commute and the numbers log N (e1), . . . , log N (en) are rationally inde-pendent. Note that the operators L∗

i were denoted by Rei in op. cit.Since X satisfies all the conditions of Remark 1.8, the algebras NOX , OX and

NO(X) coincide and therefore the next proposition follows immediately from Corol-lary 5.3.

Proposition 11.3. In the setting of the previous proposition, for every β ∈ R, we havean affine one-to-one correspondence between the gauge-invariant σ -KMSβ -states onNOX and the probability measures μ on Z such that

N (ei )−βL∗

i μ = μ for i = 1, . . . , n.

It isworth stressing that for n ≥ 2 it is not enough for aσ -KMSβ -stateφ onNT (X) tobe of infinite type to factor throughNOX , since infiniteness onlymeans thatμ = φ|C(Z)

is in the kernel of the operator∏n

i=1(1 − N (ei )−βL∗i ). The difference is particularly

illuminating in terms of the decomposition φ = ∑F⊂{1,...,n} φF from Sect. 10: φ is

infinite if and only if φ{1,...,n} = 0, while φ factors throughNOX if and only if φF = 0for all F �= ∅. See also [17] for a relation between the components φF and intermediatequotients of NT (X).

Let us now give an example showing that the system of inequalities in (3.2) cannotbe replaced by a proper subsystem, which was promised in Sect. 3.

Example 11.4. Consider the space Y = {0, 1}N. Let h : Y → Y be the shift map and ν bethe Bernoulli measure on Y with weights (1/2, 1/2). Then for the dual Ruelle transferoperator L∗ we have L∗ν = 2ν.

Fix n ≥ 2 and a nonempty subset I ⊂ {1, . . . , n}. Denote by Z the disjoint union oftwo copies of Y and define commuting surjective local homeomorphisms hi : Z → Z ,1 ≤ i ≤ n, as follows. If i ∈ I c, then hi preserves each copy of Y and coincides withh on each of them. If i ∈ I , then hi is the composition of the local homeomorphism


as in the case i ∈ I c with the flip homeomorphism exchanging the two copies of Y .The corresponding dual Ruelle transfer operators L∗

i leave the two-dimensional spaceV spanned by the measures (ν, 0) and (0, ν) invariant, and in the basis given by thesetwo measures the restrictions of L∗

i to this subspace are represented by the matrices

L∗i |V =

(2 00 2

)

, if i ∈ I c, and L∗i |V = S =

(0 22 0

)

, if i ∈ I.

Now, fixα > 2 and define a homomorphism N : Zn+ → (0,+∞) by letting N (ei ) = 2

for i ∈ I c and N (ei ) = α for i ∈ I . Put β = 1; note that this is precisely the criticalinverse temperature βc if I �= {1, . . . , n}, and β > βc = log 2/ logα if I = {1, . . . , n}.Consider the measure

μ = (1 − α−1S)1−|I |(0, ν).

In the basis {(ν, 0), (0, ν)} of V it is represented by the vector

(1 − 4α−2)1−|I |(

1 2α−1

2α−1 1

)|I |−1 (01

)

,

so it is positive. We claim thatμ satisfies (11.1) for all nonempty subsets J ⊂ {1, . . . , n}different from I and it does not satisfy (11.1) for J = I .

Let us start with the case J �= I . Consider two subcases. Assume first that J �⊂ I .Then, for any j ∈ J\I , the operator 1 − N (e j )−1L∗

j = 1 − 2−1L∗j is zero on V , so

∏

i∈J

(1 − N (ei )−1L∗

i )μ = 0.

Assume now that J � I . Then the measure∏

i∈J

(1 − N (ei )−1L∗

i )μ = (1 − α−1S)|J |μ = (1 − α−1S)1+|J |−|I |(0, ν)

is represented by the vector

(1 − 4α−2)1+|J |−|I |(

1 2α−1

2α−1 1

)|I |−|J |−1 (01

)

,

so it is positive.On the other hand, for J = I the measure

∏

i∈I(1 − N (ei )

−1L∗i )μ = (1 − α−1S)|I |μ = (1 − α−1S)(0, ν)

is represented by the vector(

1 −2α−1

−2α−1 1

) (01

)

=(−2α−1

1

)

,

so it is not positive, proving our claim.If I �= {1, . . . , n}, then, for every β ≤ βc = 1, it is actually not difficult to describe

explicitly all probability measures μ on Z satisfying (11.1) for all nonempty J . The keypoint is that Ruelle’s Perron-Frobenius theorem (see [2, Theorem 1.7]) implies that if a


probability measure η on Y satisfies L∗η ≤ c η for some c ≤ 2, then c = 2 and η = ν.Therefore, if a probability measure μ on Z satisfies (11.1) for some β ≤ 1 and J = { j},with j ∈ I c, then β = 1 and μ = (λν, (1 − λ)ν) for some λ ∈ [0, 1]. By the abovearguments, such a measure satisfies (11.1) for β = 1 and all nonempty J if and only if(1 − α−1S)|I |μ is a positive measure, that is, the vector

(1 −2α−1

−2α−1 1

)|I | (λ

1 − λ

)

has nonnegative entries. A simple computation shows that this means that

∣∣∣λ − 1

2

∣∣∣ ≤ 1

2

(α − 2

α + 2

)|I |.

Note also that Proposition 11.1(ii) gives a description of measures satisfying (11.1)for all nonempty J and any fixed β > βc, but it is not very explicit.

This example can be modified in several ways to satisfy additional properties. Forinstance, we can replace hi by h2i for all i ∈ I c, let N (ei ) = 4 for such i’s and takeα = 4, thus getting N (ei ) = 4 for all i = 1, . . . , n. We can also take the disjointunion of Z with another compact Hausdorff space W equipped with n commutingsurjective local homeomorphisms and consider a nontrivial convex combination of themeasure μ defined above with a suitable measure on W in order to get a probabilitymeasure η on W � Z such that the measures

∏i∈J (1 − N (ei )−1L∗

i )η are positive andnonzero for all nonempty subsets J ⊂ {1, . . . , n} different from I , while the measure∏

i∈I (1 − N (ei )−1L∗i )η is not positive.

11.2. Higher rank graph C∗-algebras. Let k ≥ 1. Suppose that (�, d) is a k-graph withvertex set �0 in the sense of [18].

For n ∈ Zk+, we write �n = {λ ∈ �∗ : d(λ) = n}. A k-graph is finite if �n is finite

for all n ∈ Zk+. Given v,w ∈ �0, v�nw denotes {λ ∈ �n : r(λ) = v and s(λ) = w}.

We say that � is row finite if v�n is finite all n ∈ Zk+ and v ∈ �0. The k-graph � has

no sources if v�m �= ∅ for every v ∈ �0 and m ∈ Zk+. For μ, ν ∈ �, we write

�min(μ, ν) = {(ξ, η) ∈ � × � : μξ = νη and d(μξ) = d(μ) ∨ d(ν)}.We say that � is finitely aligned if �min(μ, ν) is finite for all μ, ν ∈ �.

Following [29], for a finitely aligned k-graph �, a Toeplitz-Cuntz-Krieger �-familyin a C∗-algebra B is a set of partial isometries {Tλ : λ ∈ �} such that(1) {Tv : v ∈ �0} is a collection of mutually orthogonal projections,(2) TλTμ = Tλμ if s(λ) = r(μ),(3) T ∗

μTν = ∑(ξ,η)∈�min(μ,ν) TξT ∗

η for all μ, ν ∈ �.

The Toeplitz algebra T C∗(�) is generated by a universal Toeplitz-Cuntz-Krieger �-family {tλ : λ ∈ �}.

Given a finitely aligned k-graph, [29, Corrolary 7.5] shows that T C∗(�) is the Nica-Toeplitz algebra of a compactly aligned product system X (�) of C∗-correspondencesover Z

k+ constructed as follows. The coefficient algebra is C0(�

0). For each n ∈ Zk+, the

set Cc(�n) = span{δλ : λ ∈ �n} is a right C0(�

0)-module with the module structure(ξ · a)(λ) = ξ(λ)a(s(λ)). Then X (�)n is the completion of Cc(�

n) in the norm arising


from the C0(�0)-valued inner product 〈ξ, ζ 〉(v) = ∑

μ∈�nv ξ(μ)ζ(μ). Now X (�)n

becomes a C∗-correspondence over C0(�0), with the left action given by (a · ξ)(λ) =

a(r(λ))ξ(λ). Themultiplication in X (�) is given by δλδμ = δλμ if r(μ) = s(λ) and zerootherwise (see [29, Proposition 3.2]). The isomorphism ω : T C∗(�) → NT (X (�)) isgiven by ω(tλ) = iX (�)(δλ).

Proposition 11.5. Let � be a finitely aligned k-graph. For each 1 ≤ i ≤ k, let Ai ∈M�0(Z+) be the matrix with entries Ai (v,w) = |v�ei w|. Consider a homomorphismN : Z

k+ → (0,+∞) and the dynamics σ on T C∗(�) defined by σt (tλ) = N (e j )i t tλ for

λ ∈ �e j . For every state φ on T C∗(�) define a vector Vφ = (φ(tv))v∈�0 . Then

(i) for every β ∈ R, the map φ �→ Vφ defines a one-to-one correspondence between

the gauge-invariant σ -KMSβ -states on T C∗(�) and the vectors V ∈ [0, 1]�0such

that ‖V ‖1 = 1 and∏

i∈J

(1 − N (ei )−β Ai )V ≥ 0

for every nonempty subset J ⊂ {1, . . . , k}.Assume in addition that � is finite, N (ei ) > 1 for all i , and put

βc = max1≤i≤n

log r(Ai )

log N (ei ),

where r(Ai ) is the spectral radius of the matrix Ai . Then

(ii) for everyβ > βc, there is an affine bijection between the σ -KMSβ -states onT C∗(�)

and the set

�β = {W : W ∈ [0, 1]�0

and∥∥

∑

n∈Zk+

N (p)−β AnW∥∥1 = 1

},

where An = An11 . . . Ank

k , or equivalently, An(v,w) = |v�nw|.Proof. Working with NT (X (�)) instead of T C∗(�), let us first of all compute theoperators Fn on C0(�

0)∗ = �1(�0).A straightforward computation using the formulas for the right action and the inner

product for the module X (�)n shows that the set {δλ : λ ∈ �n} forms a Parseval framefor X (�)n . Hence, for each v ∈ �0, applying (1.10) we get

Trnτ (δv) =∑

λ∈�n

τ(〈δλ, δv · δλ〉).

Since δv · δλ = δv,r(λ)δλ and 〈δλ, δλ〉 = δs(λ), we see that the above expression equals∑

λ∈v�n

τ(〈δλ, δλ〉) =∑

λ∈v�n

τ(δs(λ)) =∑

w∈�0

|v�nw|τ(δw).

In other words, if V = (τ (δv))v∈�0 is the vector defined by τ , then the vector definedby Fnτ is AnV .

Similarly to Proposition 11.1, the result now follows fromCorollary 3.3, identity (7.8)and Corollary 7.5. ��


For finite k-graphs part (i) of the above proposition recovers [7, Proposition 4.3] thatwas proved using the groupoid picture. Together with the theory developed in Sect. 10,leading to Example 10.6, this provides an alternative route to the classification of gauge-invariant KMS-states in [7, Theorem 5.9]. Part (ii) recovers [15, Theorem 6.1(a)].

Following [18], for a row finite k-graph � with no sources, we say that a Toeplitz-Cuntz-Krieger �-family in a C∗-algebra B is a Cuntz-Krieger �-family if it satisfies

(4) Tv = ∑λ∈v�n TλT ∗

λ for all v ∈ �0 and n ∈ Zk+.

TheCuntz-Krieger algebraC∗(�) is the quotient ofT C∗(�)by the ideal⟨tv−∑

λ∈v�n tλt∗λ :v ∈ �0

⟩. By [29, Corollary 4.4], C∗(�) coincides with OX (�). Since � is row finite

and has no sources, the left action C0(�) on each fiber is by compact operators and isinjective [12]. Thus, by Remark 1.8,C∗(�) is the same as NO(X (�)). By Corollary 5.3we now get the following result.

Proposition 11.6. Let � be a row finite k-graph with no sources, N : Zk+ → (0,+∞) be

a homomorphism and σ be the dynamics on C∗(�) defined by σt (tλ) = N (e j )i t tλ forλ ∈ �e j . Then, for every β ∈ R, we have an affine one-to-one correspondence betweenthe gauge-invariant σ -KMSβ -states on C∗(�) and the vectors V ∈ [0, 1]�0

such that‖V ‖1 = 1 and

N (ei )−β AiV = V for i = 1, . . . , k.

For finite k-graphs with no sources this recovers [6, Theorem 7.4].

Acknowledgements Open Access funding provided by Oslo University and Oslo University Hospital. Thisresearch was initiated when Z.A. visited the Department of Mathematics at the University of Oslo. She thanksher colleagues for their hospitality.

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